Keita has only one year left on his contract with Lazio and has been pursued by a number of sides including Juventus and Spurs, who are supposedly readying a €30.2m (£28m) bid for the versatile attacker.
The 22-year-old was widely expected to leave the Stadio Olimpico this summer, but Tare seems to have other ideas as he seeks a solution that benefits both club and player.
Tare hinted at his frustration with Keita's agent and, while he admits there are a number of "important" teams monitoring the Senegal international, he has called on the Lazio star to be "grateful" for what the club has done for him.
"He's a Lazio player. There are important European and non-European teams that want him," Tare told Sky Sport Italia, relayed by Football365. "But at the moment, there's a war of attrition, especially on his agent's part.
"Still, we have so much confidence and patience. Every decision will be made for everyone's benefit, including the player, because no-one comes out of this as the winner.
"Therefore, I'm using common sense. I hope to find a solution, not only for the player but also Lazio.
"We mustn't forget that Keita's a player who came to Italy when he was 15 and our club have given him so much. I think he must take that into account and be grateful.
"At the same time, we're also aware that he's grown a lot and he has high expectations, but a solution must be found for the good of everyone.
"If he wants to extend his contract, he knows what the club want to do. But at the same time, if there are deals that satisfy Lazio, we're willing to take them into consideration."
Tottenham's lack of activity earlier in the transfer window was well publicised, but the wheels have started turning for Mauricio Pochettino's men in recent weeks.
Tottenham recently completed the signing of Davinson Sanchez for a club-record fee of £42m, while Pochettino also welcomed former Southampton cohort Paulo Gazzanigga to the club earlier this week.
Along with Sanchez and Gazzanigga, Tottenham are close to completing the signing of full-back Serge Aurier for around £23m. Aurier has been usurped at Paris Saint-Germain by Dani Alves and Thomas Meunier, and recent reports suggest that a deal to bring him to north London is nearly finalised.