The Premier League duo will have to pay £3m more than the Saints, with whom the Belgian defender spent the season on loan from Atletico Madrid due to a clause in the contract the two clubs had agreed at the start of the season, according to the Telegraph.
Under the contract, Atletico will have to pay Southampton a compensation of £3m if Alderweireld decides to quit the south coast club for one of their rivals in the Premier League.
The Belgian defender, who was not part of Diego Simeone's plans at the Spanish capital club, has spent a successful year in England with the Saints and his performances have attracted the interest of the big guns, who are looking for defensive reinforcements ahead of next season.
Mauricio Pochettino has already indicated that he is looking to sign players with Premier League experience and wants defensive cover as they are set to offload Younes Kaboul and Vlad Chiriches during the summer transfer window.
The Blues boss Jose Mourinho has also admitted that he is looking to add one or two players to his squad in the off-season, but made it clear that he is not looking to spend big and is only keen on players who can provide backup and challenge his first-team stars during the course of the campaign.
Alderweireld is a valuable asset as he can play all across the back four, either as a centre-back or in the full-back positions. The Belgian has confessed his desire to remain in England, after enjoying a successful spell at St Mary's, but he is unclear if his future lies with Southampton or one of their rivals in the league.
"There are three options – stay with Atlético, stay with Southampton or go to another club. Southampton gave me that opportunity and it paid off, I think. I've had a fantastic year, I can't lie about that, so, yes, I'd like to stay in the Premier League," Alderweireld is quoted as saying by the Telegraph.