The Lilywhites are expected to be in the market for a new number one this summer to succeed Hugo Lloris, who is out of contract at the end of next season.
Lloris - who is currently sidelined with a knee injury - has also been criticised for a decline in performances this season, and the former France captain turns 37 in December.
Despite his Arsenal connections, World Cup Golden Glove winner Martinez has seemingly emerged as a prime target for Spurs amid suggestions that Villa are willing to cash in this summer.
However, speaking after the weekend's 3-0 win over Bournemouth - where he equalled Villa's record for the most clean sheets in a goalkeeper's first 100 Premier League games for the club - Martinez affirmed that he sees his long-term future in the West Midlands.
"My career went through the roof since I joined the club. I've always said I'm grateful to this club and I love being here," the 30-year-old told the press.
"My family is happy at the club, my boy is obsessed with playing football and he plays for the under-6s in the academy. We're loving our time here and hopefully there are many years to come."
Martinez's shut-out against Bournemouth was his 33rd clean sheet in 99 top-flight games for Villa, the same number achieved by Brad Friedel and Mark Bosnich in 100 league appearances.
The former Arsenal man largely featured as an understudy during a 10-year spell at the Emirates, only making his breakthrough in 2020 due to Bernd Leno's injury and starring as the Gunners won the FA Cup.
The World Cup winner could break Villa's 100-game clean sheet record when the Lions meet Chelsea in their first contest back following the international break, as Tottenham search elsewhere for a new number one.
Everton's Jordan Pickford was believed to have been a prime target before he signed a new Toffees deal, but he could still be allowed to leave for a respectable fee if Sean Dyche's team are relegated from the top flight.
Another former Arsenal goalkeeper in Wojciech Szczesny has supposedly turned down a move to the Gunners' bitter rivals, but Dean Henderson and Giorgi Mamardashvili are also admired by Tottenham chiefs.
Fraser Forster's form during Lloris's injury has earned the 35-year-old a recall to the England squad in place of Nick Pope, but Brandon Austin and Alfie Whiteman only have one senior Spurs appearance between them.
Tottenham's summer transfer window is highly unlikely to be overseen by current manager Antonio Conte, who is now expected to be sacked in the wake of his scathing assessment of his "selfish" players at the weekend.
The Italian ripped into his squad after they threw away a 3-1 lead to draw 3-3 with Southampton on Saturday, and Ryan Mason - who also stepped in after Jose Mourinho's sacking in 2021 - could take charge until the end of the season following the international break.