Brendan Rodgers had hoped to have the move wrapped up before the weekend but still no confirmation has come.
English journalist Johnson covers PSG in France and understands Weah is poised to become the Hoops' second January arrival after Oliver Burke joined on loan.
"I think absolutely it will go through," Johnson told the Transfer Talk podcast.
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"He needs as much gametime as possible and PSG can't offer him that and Celtic are a club that PSG have had good experiences with in the past.
"They sent Odsonne Edouard there last season, he had a great campaign and they ended up buying him permanently after that.
"I don't think PSG will be willing to part with Weah at this stage of his career but they are desperate for him to get game-time - as is he.
"It's a move that makes sense for all parties.
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"It will be interesting to see exactly when a final decision is made."
Weah has eight caps for the USA and is highly rated by PSG boss Thomas Tuchel.
Rodgers is in the hunt for a frontman as he Edouard is the club's only recognised centre-forward.
That follows Leigh Griffiths' decision to take time away from the game to deal with personal problems.