The highly-rated Argentine was heavily tipped for the Old Trafford job when Jose Mourinho was sacked in December.
Even when Solskjaer was appointed interim boss until the end of the season, Pochettino was still believed to be among the frontrunners for the permanent role in the summer.
However, Solskjaer’s impressive three-month spell in interim charge, which saw the Red Devils win 14 out of 19 games in all competitions, led to the Norwegian being handed a permanent three-year contract on Thursday.
When asked about the job, Pochettino refused to be drawn on whether he really was a contender for the role, but did have some kind words to say about the new permanent United boss.
"I never talked about rumours," said Pochettino. "I need to congratulate him and wish him all the best, now he is officially the new manager of Manchester United.
"I am not going to talk about rumours and in this period there is going to be more rumours, but there is nothing I can do about it.
"If it is not this, speculation is going to come around from another club. We are so focused on our job."
Manchester City boss Pep Guardiola also had some kind words to say about Solskjaer, now the permanent boss of his club’s bitter rivals.
“United are always an incredible opponent, congratulations to Solskjaer, he has done incredible in terms of results and I wish him all the best,” said Guardiola.
Solskjaer’s first game in permanent charge is against Watford at Old Trafford on Saturday.
Pochettino’s Tottenham, meanwhile, are just one point better off in third, having seen their title challenge crumble in recent weeks.
Spurs were just five points off the top at the end of last month but three defeats in four Premier League matches has seen them fall 15 points behind leaders Liverpool, who they face at Anfield on Sunday.