The allegation was made by Morocco midfielder Nordin Amrabat after a game in which American official Geiger made a number of calls that were questioned by the north Africans.
Amrabat told Dutch television station NOS: "I do not know what he is used to but he was very impressed with Cristiano Ronaldo and I hear from [Portugal defender] Pepe that he asked Ronaldo in the first half if he could have his shirt.
"What are we talking about? At the World Cup? It’s not a circus here."
FIFA responded by backing Geiger and expressing their disappointment at Amrabat's outburst.
In a statement it said: "It is with regret and disappointment that, following yesterday's Group B game between Portugal and Morocco, FIFA has learnt of reports in the media concerning the referee Mark Geiger.
"It has been claimed that Mr Geiger asked for the shirt of the captain of Portugal during the half-time interval.
"Mr Geiger strongly refutes those claims and categorically states that such a request was not made.
"FIFA unequivocally condemns the allegations supposedly made by a member of the Moroccan team.
"FIFA referees are under clear instructions with regard to their behaviour and relationship with the teams at the World Cup in Russia and it can be confirmed that Mr Geiger has acted in an exemplary and professional manner as an appointed match official.
"FIFA would like to remind teams of their duty to respect all principles of Fair Play."
Morocco coach Herve Renard singled out Geiger in his post-match press conference, although stopped short of criticising him directly.
However, it was clear the Frenchman thought Ronaldo's fourth-minute header, which turned out to be the only goal, should have been ruled out while Geiger also turned away a number of confident penalty appeals from Morocco.
Renard said: "I'd like you [the media] to be honest. Have a good look at the corner when Portugal scored and have a good look at what their number three is doing and write the truth.
"It's not up to me to comment on refereeing, I would ask you to have a look at the replays, the slow-motion clips, because a foul should have been seen and there should have been a free-kick.
"It was missed and Ronaldo appears. If I criticise referees I will be punished so the only thing you should write is we are very proud of our performance."
The defeat meant Morocco became the first team to be eliminated from World Cup 2018.