Both players are out of contract at the end of the season, along with midfielder Christian Eriksen, and will be free to discuss terms with other clubs in January.
But Sky Sports reports that Vertonghen is eager to open discussions over a new contract now that Mourinho has replaced Mauricio Pochettino as Spurs boss.
The Belgian was also open to extending his deal under the previous management and was told that he wasn’t for sale over the summer, when Roma and Bayer Leverkusen showed interest.
However, the centre-back’s representatives are yet to sit down with the Tottenham hierarchy for formal contract talks.
Alderweireld spoke to TalkSport after Tottenham’s 4-2 comeback win over Olympiacos on Tuesday.
When asked about Mourinho having shown interest in signing him for Manchester United in the past, the defender said: “I don’t know what to say about that!
“I feel lucky to work for him, so I’ll just do my best like I always do.”
Vertonghen, meanwhile, is one of the London club’s longest-serving players, having joined from Ajax in 2012.
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