The Portugese coach is odds-on favourite with SkyBet after significant backing on Thursday evening, with White Hart Lane chiefs understood to have made initial contact.
Villas-Boas was sacked by Chelsea in February but is thought to be keen on a return to London.
Everton manager David Moyes and Wigan's Roberto Martinez are both strong candidates to replace Redknapp, who was sacked on Wednesday night, but both would bring compensation costs.
Latics chairman Dave Whelan has already said he will not stand in Martinez's way, should Spurs make an approach.
"If Tottenham come to me and ask to speak to him, I have to say the answer is 'yes'. It always is," Whelan said.
"You cannot stop managers moving. Every club that people regard as a big club, when these jobs come up, people say 'oh Roberto must be in for that'.
"But I have not heard one thing from Tottenham Hotspur or from Roberto, so as far as I am concerned there is nothing going on at present."
Manchester City will try and win back the Premier League title with this new soccer jersey from Nike!
The USA will wear their blue away jersey against Jamaica in their World Cup qualifier on Friday night.
Support the USA national team with an official 2013 Centennial soccer jersey from Nike. Its Official.
Germany is one of the top teams ever and they will play the US on Sunday.
Jamaica is in need of a win against the U.S. Get the jersey of the Reggae Boyz.
Neymar's move from Santos to Barcelona is no doubt the biggest summer transfer.
The Nike Hypervenom is the latest cleat silo from Nike. It will change your game and is worn by the game's top players.
The adidas Nitrocharge is the most talked about new cleat in years. It was released on May 15 and will take your pitch by storm.
The hoop soccer jersey is classic and cool. The USA hooped jersey is perfect for every fan. Get in the Gear.
Barcelona will wear this new home soccer jersey next season. They also have a new away jersey. Get them both.