The duo are in the midst of their own respective transfer sagas, with Martial having been informed that he can leave Old Trafford on the right terms, whereas Alderweireld has fallen out of favour in north London.
Both United and Spurs have been linked with the defender and forward respectively, and Bridge claims that the Red Devils could be willing to pull off their second swap deal of the year after Alexis Sanchez and Henrikh Mkhitaryan switched allegiances in January.
"Anthony Martial's future could be the big talking point of the transfer window," Bridge told Sky Sports' Transfer Talk podcast, via the Express.
"Chelsea have entered the fray and I have said plenty of times that Spurs have a longstanding interest in Martial and I still stand by that.
"There may even be a deadline day swap between him and Toby Alderweireld, who knows. There will still be plenty to go in the last few weeks.
"Chelsea, we know they are interested but they are yet to make a formal offer. Sky in Germany are saying Bayern Munich will only make a move if United lower their valuation of the player - you'd say unlikely.
"Anthony Martial's agent has said Martial is not wanted as much at Manchester United but Jose Mourinho has said a lot of his World Cup players won't be starting the season.
"He's actually started for United against Club America and San Jose Earthquakes. Jose Mourinho has said he wants to start Martial against Leicester alongside Sanchez and Juan Mata.
"Is it Mourinho and Man Utd not appreciating him? We don't know. It might have taken them by surprise that Martial has said he wants to move on. not many players want to move on from Manchester United," he added.
Martial has also received interest from Chelsea but United have remained insistent that they will not sanction a deal to a league rival, while Spurs are willing to sanction a deal for Alderweireld with a domestic rival as long as their valuation for the defender is met.