The 24 year-old trained with Pompey yesterday (July 27) and played 27 minutes of a 2-0 friendly defeat at Chelsea Under 23s.
Pompey boss Danny Cowley told the Portsmouth News: “He is training with us this week and we really appreciate Peterborough allowing that to happen.
“It’s a great opportunity for him to look at us and it’s an opportunity for us to look at him, and then we’ll make a decision on the back of it.
“Dan trained this morning and had half an hour this evening in the match.
“He’s a really good character. He’s a good size, has a good presence with his physicality and is a boy that loves his football.
“I often think being number two or number three goalkeeper is the hardest role in the squad. They always have to be really, really good characters.”
Gyollai has yet to make a Football League outing, although appeared four times in last season’s EFL Trophy for Posh.