The duo, who won the Australian Open men's doubles title in January, were beaten 6-7 (2-7) 7-6 (7-2) 10-8 by Feliciano Lopez and Marc Lopez.
Scot Murray, 30, is looking to overtake Soares' fellow Brazilian Marcelo Melo.
"It didn't happen today, but next week we'll go again," said Murray.
"I was aware of it. People write about it so you hear about it.
"It probably came into my head a couple of times. I guess that's difficult to block that out. It's not every day you get the chance to reach the top in your sport and for me it was a huge match."
Doubles players are ranked individually and Murray, who helped Britain win the Davis Cup in November, has had 63 playing partners since 2003.
Melo will have to reach at least the semi-finals in Miami, which starts next Wednesday, to deny Murray top spot in the rankings.
Jamie Murray will have to wait to become the world number one in doubles after he and partner Bruno Soares lost in the quarter-finals of the BNP Paribas Open in Indian Wells.