“I definitely think I need to be concerned about today,” Murray said.
“It’s not always the worst thing losing a match, but it’s sometimes the manner of how you lose the match which can be concerning or disappointing.”
Coric, 20, only gained a place at the tournament after Richard Gasquet withdrew – becoming the first lucky loser to reach the quarter-finals in the Madrid tournament’s 16-year history.
The Croat broke his Scottish opponent three times in the opening set, and a further break in the second was enough to secure victory in one hour and 25 minutes.
Top seed Murray hit 14 winners to his 28 unforced errors, but insisted his poor performance was not down to a lack of confidence.
“I was just making lots of mistakes early in the rallies and trying to end points very quickly at the beginning, and the errors just kept piling up.” the 29-year-old told BBC Sport.
“I didn’t feel that was down to confidence – I just wasn’t focusing as well as I needed to on each point.
“I made a lot of unforced errors and I also didn’t find any way to make it a more competitive match, so that’s the most disappointing thing for me.
“You can lose matches sometimes, but the manner of today’s loss was disappointing.”
BBC tennis correspondent Russell Fuller:
This result will come as a shock to Murray’s system.
He had seemingly been growing in confidence, and rediscovering his rhythm little by little as he made his way from Monte Carlo to Barcelona, but now has just one week in Rome to find the form and belief which would make him a genuine contender for the French Open.
His first serve, which has been hindered by an elbow injury, was not to blame against Coric, who played aggressively and fluently and took full advantage of Murray’s error-strewn performance.
Her involvement in Spain, courtesy of another wildcard, ended in the second round courtesy of a 7-5 2-6 6-4 defeat to the Canadian, who had been critical of the Russian, calling her a “cheat” and saying she should never have been allowed to play again.
She also received a wildcard for Stuttgart in April, where she reached the last four.
The line-up for Birmingham is the strongest ever seen at the grass-court event.
Of the current top 10, only Svetlana Kuznetsova and Serena Williams, who recently announced her pregnancy, are missing.
The event runs from 17-25 June and is one of three tournaments – along with the Aegon Open in Nottingham and the Aegon International in Eastbourne – in the lead up to Wimbledon in July.
Sharapova, who has not played a Wimbledon warm-up event since she reached the final in Birmingham in 2010, remains some way adrift of direct entry into the Wimbledon main draw.
She will need to reach the semi-finals in Rome next week to make sure. A first-round defeat could cost her a place in qualifying unless the All England Club offers her a wildcard.
Maria Sharapova will be offered a wildcard into next month’s Aegon Classic in Birmingham.
Sources told BBC Sport that preliminary discussions between Sharapova’s team and the Lawn Tennis Association are under way.
The move has the blessing of the LTA’s outgoing chief executive Michael Downey.
Sharapova has not played a Wimbledon warm-up event since she reached the final in Birmingham in 2010.
Her presence in the field is likely to boost ticket sales at an event that is perpetually overshadowed by the ATP tournament at The Queen’s Club in the same week. Tickets are still available for all days of the tournament which runs from 19-25 June.
The LTA will expect criticism for their decision, which appears to have been made purely on commercial grounds. Andy Murray and Heather Watson are just two players who have expressed their uneasiness about the quantity of wildcards being offered to Sharapova since she returned from a 15-month ban for taking meldonium.
Watson, who will receive a wildcard of her own into the Birmingham draw, said during March’s Miami Open that from “a moral standpoint you should have to work your way back up if you’ve been on a ban”.
There is unlikely to be swift confirmation that Sharapova will play in Birmingham, where she won the title in both 2004 and 2005. That is because all eyes are currently on Roland Garros, which will announce next Tuesday whether they are to offer Sharapova a wildcard for either qualifying or the main draw of the French Open.
That, in turn, will have implications for how much Sharapova wishes to play on the grass. The five-time Grand Slam champion has already been offered a wildcard into the event in the Dutch town of Rosmalen in the week after the French Open.
Sharapova’s participation in the rival Nottingham tournament should not be completely ruled out if she loses early in Paris, or a French Open invitation fails to materialise.
The All England Club will not meet to finalise their wildcards until 20 June, but despite her second-round defeat to Eugenie Bouchard in Madrid, Sharapova can still gain direct entry into Wimbledon by reaching the semi-finals in Rome next week.
Johanna Konta, Garbine Muguruza, Agnieszka Radwanska and Simona Halep headline the entry list in Birmingham, which will be confirmed later this week.
“We are currently in the process of deciding which players will receive wildcards for our summer season grass court tournaments” an LTA spokesman told the BBC.
“It’s a carefully considered process which happens at this time every year. We will announce the names of wild card recipients in due course, before the start of each tournament.”
World number one Andy Murray will defend his title at next month’s Aegon Championships as six of world’s top 10 men descend on Queen’s Club in London.
The Briton has won the traditional Wimbledon warm-up event five times.
The tournament will also feature last year’s beaten finalist Milos Raonic, 2012 champion Marin Cilic and 2008 winner Rafael Nadal.
Stan Wawrinka and David Goffin are also scheduled to play in the event which takes place from 19-25 June.
Jo-Wilfried Tsonga, Tomas Berdych, Nick Kyrgios and Juan Martin del Potro will also take part.
Tournament director Stephen Farrow said: “The player entry list for the Aegon Championships is strong every year and we already knew we were in for a great line-up, but with Cilic and Goffin adding their names to make it six of the world’s top 10, this is going to be the best yet.”
“I don’t think he really believes he’s British either,” said Evans after losing in the Madrid Open first round to Robin Haase on Monday.
“It’s nothing against Aljaz. I like him, he’s not confrontational in any way – but to me it doesn’t sit well if you play for another country.
“I don’t feel bad about him, but for me it’s a bit baffling as to why.”
The BBC has contacted Bedene for his response to Evans’ comments.
It is not the first time Evans has spoken out against Bedene’s switch, with the player previously saying – before the change of allegiance was confirmed – Bedene should not be allowed to represent Britain in the Davis Cup.
International Tennis Federation rules do not allow players to represent more than one country in the Davis Cup.