The club paid £58million to sign the American sensation, 20, from Borussia Dortmund.
But he will not arrive at Stamford Bridge until the end of the season as he has been loaned back to Dortmund for the rest of the current campaign.
Sarri watched his Chelsea side labour to a 0-0 draw at home to Southampton on Wednesday night and winger Willian came off injured to compound their woes.
Pedro is also out injured but the Spaniard is nearing a return to action.
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Pulisic could have come in as a direct replacement for either player but Sarri claims he had no say on the finer details of the deal.
Asked if the club should have brought him in immediately, he replied: “I don’t know.
“As I said before I am not in charge of the market, it is difficult for me to talk about the market or the deal of Pulisic.
“Willian’s injury I think is not very serious. The doctor said it was not serious for Pedro as well.
“Pedro in one week can train. Of course this match was difficult as Pedro, (Callum Hudson-) Odoi, (Olivier) Giroud were out.
“On the bench I had one offensive player, (Ruben) Loftus (-Cheek) is not at 100%, the bench wasn’t offensive so it was difficult to change the match.
“I put on (Cesc) Fabregas as in that moment of the match we didn’t move it like we did in the first half. It was really very difficult for me to change.”
Sarri hinted that he hoped the board would look to continue spending in the January transfer window to help as looks to strengthen his squad.
It is claimed he is looking to bring in another striker, with Gonzalo Higuain and Callum Wilson reported to be targeted.
He admits his side have prelims in attack as he watched Alvaro Morata fire yet another blank in front of goal.
When asked if he would look to solve those problems by going back into the market he claimed transfers were out of his control.
“I have to think about the pitch, improving my team and the offensive phase,” the Italian added.
“We are good to build up the action, we have to improve in the last 20 metres. The club know my position, it is up to the board.”