Inter believe a total package of £30m could be enough to get a deal done and Spurs are understood to be willing to sell if they receive an acceptable offer.
Spence's opportunities could be limited at Spurs this season with Destiny Udogie and Andy Robertson also competing for the left-back position and the ever-present Pedro Porro the clear first choice on the right.
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Inter want to bring in a new player who can operate as a wing-back.
Spence has been high on their list all summer and Inter are now turning their attention to the Spurs man after struggling to strike a deal for Al Ittihad's Moussa Diaby.
De Zerbi: Spurs aren't done in the transfer window
Roberto de Zerbi maintains Tottenham are "not finished yet" in the transfer market as they look to build a competitive Premier League squad again.
Having only avoided relegation on the final day of last season, Spurs have spent already spent around £237m - with the club's transfer record broken twice in deals for Mateus Fernandes and Sandro Tonali.
Dutch defender Jan Paul van Hecke also arrived from Brighton for £52m, while Martin Dubravka, Robertson and Marcos Senesi signed on as free agents.
With Micky van de Ven having also committed his long-term future to the club, there is genuine hope Spurs will soon be challenging at the other end of the table.
"In the transfer market, we have not finished yet [with] the players who will arrive," De Zerbi said, speaking to the Men in Blazers Network.
"We are looking forward to start another season with, I don't know [different] ambitions, but with different results."
De Zerbi continued: "The club has been great because they closed [deals for] very, very important players. I think we have not finished yet.
"Okay, we spent money, but the other clubs made the same or more in the past. To compete in the Premier League, you need first [the] level of the players.
"The ambition of the owner of the club, of the board, of the people who decide behind me are very important and very high, so we have to find the condition to build a strong team, a stronger team than the last seasons, to compete again for the top of the table."
