Berahino was the subject of interest from Spurs in the summer but West Brom rejected their offers for the player.
Berahino handed in a transfer request and threatened not to feature for the Baggies again after West Brom prevented him from making the move.
The 22-year-old eventually made his return back into the side scoring just three goals this season in 11 appearances.
Chelsea are keen on bolstering their attacking options with Diego Costa struggling to recapture his form of last season and both Radamel Falcao and Loic Remy struggling with fitness issues.
Jose Mourinho last week insisted he would not ask for more players in January but conceded he would take them if the club wanted to make moves.
United’s interest in Berahino remains a speculative one having already signed Anthony Martial in the summer and using Memphis Depay as an auxiliary striker to good effect against Watford.
Tottenham failed to sign a back up for Harry Kane and have started the season strongly with a number of young English players in the side.
Mauricio Pochettino refused comment on the Berahino situation in the summer, but Spurs made a statement on their interest insisting they had acted in the correct manor.
West Brom chairman Jeremy Peace had claimed Tottenham had unsettled their player and said they would refuse all offers.
West Brom sit at 12th after 13 games and Berahino is the club’s joint top scorer with James Morrison and summer signing Salomon Rondon with three goals each.