But Bublik, 24, proved too strong for the former world number one on Sunday, winning 7-6 (11-9) 6-3 in California.

"I just didn't take my opportunities today, and I feel that was an opportunity missed," said Murray.

The first set went with serve but Murray, currently ranked 88th, spurned three break points when leading 2-1 up, and another at 3-2 up.

Then in the tie-breaker, Bublik prevailed after saving three set points and Murray failed to take another two break points in the first game of the second set.

Bublik then really found his groove, going 4-1 up after claiming the match's only break of serve.

Murray did well to hold on to his serve to make it 4-2 and continued to fight to stay in the match, but Bublik saw out the win to reach the third round.

That also left Murray without back-to-back wins since reaching the Sydney Classic final in January.

"Against players that have big serves and are not easy to break, you need to - when you get those chances - be ruthless," added Murray, who earned a wildcard for the Masters event and enjoyed his 700th win on the ATP Tour in the first round.

"I just didn't quite play well enough in those moments today and that's the thing that obviously I want to turn around."

Analysis

BBC tennis correspondent Russell Fuller

Next stop Miami for Andy Murray, where he hopes [recently reappointed coach] Ivan Lendl will be able to watch some of his matches on site.

Murray is still in the market for a second coach, who would be prepared to travel with him for 20-25 weeks a year. He says the trial late last year with Esteban Carril went very well, but the Spaniard did not have that sort of time available.

The LTA coach Mark Hilton, who worked previously as a travelling coach with both Dan Evans and Kyle Edmund, has been with Murray in Indian Wells and will spend the next week training with him in Miami.

Murray says their time together has been both positive and enjoyable. No conversations have yet been held about whether that partnership could become more permanent, but in Murray's eyes, he seems a strong candidate.

Third seed Zverev beaten

Elsewhere, third seed Alexander Zverev is out after a three-set defeat by American Tommy Paul, who recovered from 4-2 down in the final set to seal a 6-2 4-6 7-6 (7-2) victory.

German Zverev was playing his first ATP tournament since being thrown out of the Mexican Open and given an eight-week suspended ban for striking the umpire's chair last month.

Zverev's exit means Matteo Berrettini is now the highest-ranked player in the bottom half of the draw. The Italian sixth seed beat Denmark's Holger Rune 6-4 4-6 6-4.

Andrey Rublev, seeded seventh, is also through to the third round after easing past Germany's Dominik Koepfer 7-5 6-4.