Silva, 28, has returned to Portugal following his country's Confederations Cup campaign in Russia – where they finished in third place thanks to an extra-time penalty from the midfielder which sealed a play-off victory over Mexico.
Latest reports from Portuguese publication A Bola claim the north London club have returned with a bid of €30m (£26.2m, $34m) for the player - €25m up front and an additional €5m in bonuses - having already seen an offer of €20m turned away by Sporting.
The report adds Sporting are determined to hold out for €35m, but Tottenham would appear close to tempting Sporting into doing business. Silva himself is unaware of any offer, but admits the chance to play in England would be tempting.
"I was not in touch with anyone during the Confederations Cup, I preferred to focus completely on this event and finish it in the best way," he was quoted as saying by the same publication.
Asked about a move to the Premier League, he added "The best players are there, the English championship is recognised by all as one of the best. Let's just let things run, I have a contract with Sporting, but nothing is certain."
Tottenham finished the 2016-17 season as Premier League runners up but will be expected to strengthen their squad with another Champions League campaign awaiting them next term.
In addition to Silva, Mauricio Pochettino's side also remain linked with AC Milan playmaker Suso. IBTimes UK sources understand the former Liverpool starlet is keen on making the move to Spurs after Pochettino himself identified him as his ideal target to replace Moussa Sissoko, who is expected to leave the club after one miserable campaign at White Hart Lane.
Spurs had also been linked with a move for 19-year-old Barcelona midfielder Carles Alena, who Pochettino was ready to hand a first-team role to. The Spain Under-19 international however chose to sign a new deal at the Nou Camp last week, ending Tottenham's hopes of luring him to the Premier League for now.