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.
His compatriot Alderweireld is also keen to agree fresh terms with the club, according to football.london, and Mourinho wants him to stay.
The 30-year-old is settled in London, where his partner is due to give birth to his second child soon, and would be happy to commit if he receives a suitable offer.
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.”
The defender joined Spurs from Atletico Madrid in July 2015 and has made 174 appearances for the club, including 15 this season.
Vertonghen, meanwhile, is one of the London club’s longest-serving players, having joined from Ajax in 2012.
Since then the Belgium international has made 294 appearances.
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