Speaking on the BBC’s Good Morning Scotland (GMS) programme, Mr Sarwar also said had not had enough time as leader to turn around 20 years of his party’s electoral decline.
Asked whether he was in contention for the top job, Mr Sarwar said: “Not yet, I don’t think.
“I would love to be a contender for First Minister If people choose to vote on May 6 for me to be the First Minister, of course I would be very very proud. But I’m also a realist.”
Responding to the suggestion that his outlook was defeatist, the 38-year-old said: “I am just being honest.”
“I'm being honest in saying that I don't think I can turn around the decline of the Labour Party in Scotland for the last 20 years.
“I don't think I can turn that decline around in 10 weeks - from the moment I became leader, to the moment of the election campaign.
A ‘cheap game-playing opposition’
He added that he believed he could reverse the party’s fortunes over time, but told GMS’s Gary Robertson that he was “not naive about the scale of the challenge” he faced.
Scottish Labour currently sits at around 20 per cent in the polls.
“Three days before I became leader we were at 14 per cent in the polls,” Mr Sarwar said, “we're making progress on that over the last five weeks. We still have three weeks ago until the election campaign.”
Instead, he insisted his priority was on overtaking the Scottish Conservatives to form Scotland’s main opposition party.
“On May 6 you aren't just voting for a government, you're also voting for an opposition.
“And I think it's right for us in Scotland - focusing on that recovery - to both stop and SNP majority, so they aren't blindsided by their weakness on the constitution and that one priority...and at the same time let's have an opposition that's not a cheap game-playing opposition, like we've had from the Conservatives.”