Spurs lost their Premier League opener 1-0 to Everton at Tottenham Hotspur Stadium as Dominic Calvert-Lewin headed the only goal in the 55th minute.
The reverse came days after Mourinho outlined the need for a new forward to compete with and complement Harry Kane.
Sorloth, the on-loan Trabzonspor man, is an option, according to the Turkish club's president Ahmet Agaoglu.
The Norway striker signed a temporary two-year deal with Trabzonspor last year after struggling at Crystal Palace but then scored 24 league goals in 2019-20 to attract the attention of Tottenham and RB Leipzig.
Agaoglu told Fanatik: "Sorloth wants to go. But there is no way that we will agree immediately for every player who wants to leave.
"We are meeting with Leipzig and Tottenham, ourselves and Crystal Palace. We are both entitled to this transfer, we will act in Trabzonspor's best interests."
Mourinho would not dispute the interest but was reluctant to discuss Sorloth.
"Well I'm not going to discuss Sorloth," he told reporters. "If the president of Trabzonspor said that, I'm not going to say that it's not true.
"I think Sortloth is a Trabzonspor player and it's not for me to speak about."
He added: "Do we need a striker? Yes. Was it because of that we lost the game? No. We didn't lose the game because we need a striker.
"We lost the game and the responsibility of that defeat is my responsibility."
Spurs will have to regroup quickly from the Everton loss as they face up to a packed schedule that could mean nine matches before the October international break.
As Mourinho and Tottenham dominate conversation with their documentary from last season airing on Amazon Prime, the coach is at least relieved he will no longer have to contend with such behind-the-scenes scrutiny over this stretch.
"My message is inside," he told BBC Sport. "I am so happy that there's no more Amazon, so things can stay inside between us. That's the way I like it."