Germany exited the tournament with a shootout loss to Paraguay on Monday and Morocco also sent the Netherlands packing on penalties.
"First of all I don't believe that we are the favourite team because ... the last few days (we have seen) how difficult it is for everyone," Pochettino told reporters at San Francisco Bay Area Stadium on Tuesday.
"I think favourite is (something) to talk (about) after, no? Everyone said the favourite is Germany but Paraguay beat (them) and Morocco against Netherlands.
"In this World Cup, there are many things, factors that can influence the performance. We need to be careful when we say, OK, one is favourite (over) another."
While Bosnia finished third in their group, the U.S. topped Group D by winning their first two matches against Paraguay and Australia to reach the knockout phase early.
Talisman Christian Pulisic has declared himself fit to start against the Bosnians after playing off the bench in the 3-2 defeat by Turkey but Pochettino all but ruled out centre back Mark McKenzie, who has been nursing a foot problem.
There was a better prognosis for left back Auston Trusty who came off on a stretcher near the final whistle against Turkey after rolling an ankle.
"Auston is much better than we expected. We'll make one final assessment but hopefully he'll be available on the bench," said Pochettino.
The winners will play Senegal or Belgium, who meet earlier on Wednesday.
Pochettino said the U.S. needed to regard the Bosnia game as a final and respect their opponents' quality.
"They deserve that respect," he said.
"You can see that they have quality, not only that they are aggressive and have good organization. They have players with quality and a coach that provides the platform to perform.
"That is a good thing. That is why, for us, this is the final of the World Cup tomorrow."