Express Sport understands a deal is in the pipeline and Shanley believes it took Rodgers to step in to twist the player's arm to make it happen.
"He's 18 years old and he has had talks with Brendan Rodgers and Brendan Rodgers has tried to convince him to come to Celtic," he revealed on the Transfer Talk podcast.
"Celtic have a link-up with PSG, who Weah pays for, as they sent Odsonne Edouard to Celtic last season. They had him on loan this time last year with a view to buying him.
"PSG certainly saw the benefits of Odsonne Edouard going out to Celtic and in the end they sold him to Celtic.I think there is a tie up between the two clubs.
"It was interesting to note Weah started following a lot of Celtic players on Instagram. Maybe we shouldn't read too much into that.
"But certainly there have been talks and Celtic need strikers.
Brendan Rodgers is keen to add Tim Weah to his Celtic squad (Image: GETTY)
Timothy Weah has made eight appearances for the United States national team (Image: GETTY)
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Celtic vs Rangers: Brendan Rodgers 'the best team won'
"Edouard has not always been firing this summer while Leigh Griffiths is unavailable for an indefinite period and they've sold Moussa Dembele.
"They do have issues up front and at times it does look like they lack that depth. "
Celtic have also put a new £2million take-it-or-leave it offer on the table for Vakoun Issouf Bayo.
Express Sport understands the improved bid would see the majority of the money now paid upfront and land Bayo's Slovakian club Dunajska Streda a significant future sell-on percentage.
Celtic are keen to get the transfer over the line and are hopeful of being given the green light for the twice-capped Ivory Coast international to travel to Glasgow this weekend.