The Peterborough keeper yesterday arrived on the south coast ahead of a potential switch.
After training on Tuesday morning, he played 27 minutes of the Blues’ 2-0 defeat against Chelsea under-23s.
The Posh are willing to let Gyollai depart – and are prepared to cancel the remaining 12 months of his contract to enable it.
Pompey are seeking another goalkeeper after a deal for Corey Addai last week collapsed.
Now Cowley will cast his eye over Gyollai.
Cowley told The News: ‘He is training with us this week, 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.
The 24-year-old also spent time on loan at non-league Nantwich Town during his time on the books of Stoke.
He featured for 63 minutes in the 2-0 defeat, with Gyollai replacing him for the remainder of the friendly at Cobham.
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