The PSG forward has been the subject of a late loan offer but Parkhead is his preferred destination if the French champions accept the deal, according to the Scottish Sun.
Weah, the son of Liberian legend George, has been capped eight times by the US national team and is seeking first-team football having played just 143 minutes this season.
The 18-year-old, who has netted twice this campaign, will likely complete the deal ahead of Celtic’s winter training camp in Dubai. Unlike the Odsonne Edouard deal, however, there will be no option for the club to make the deal permanent with PSG wanting the player to return in the summer.
Weah could be joined by another promising forward with Celtic expressing an interest in Brighton & Hove Albion’s Aaron Connolly as they seek two strikers in January.
The Irishman has been in stunning form for the club’s U23 side, netting 16 goals in as many appearances, which has led to links with Manchester City and Bundesliga side Hoffenheim.
Brighton are in talks with the player over a new long-term contract but could be willing to allow the player to leave on loan for the rest of the season, report the Scottish Sun.
The club have been dealt a blow, however, in their pursuit of a full-back.
It emerged earlier in the week that Atalanta full-back Timothy Castagne was a key transfer target as Brendan Rodgers looks to strengthen his defence.
Celtic were reportedly willing to pay £6million for the Belgian international’s services, yet the player has revealed that he is hoping for a move to either England or Germany.
Meanwhile, the club won’t be signing Spurs striker Vincent Janssen.
The Duthman’s name was linked to the Parkhead side heavily on social media on Thursday but even though he will likely leave in January, the £17million signing won’t be heading to Glasgow, according to the Daily Mail.
• Former Celtic youngster Ciaran McKenna is expected to join Hearts on trial.
The player left the club three years ago after rejecting a professional contract to move to the United States on a scholarship.