Manchester United had told Fernandes' camp to keep them informed of any other enquiries made, according to The Telegraph.
Miguel Pinho, Fernandes' agent, is reported to have met Spurs representatives in London this week.
The north Londoners are weighing up moves for midfielders after Christian Eriksen made it clear he wanted a new challenge.
Eriksen has been linked with a move to Real Madrid, but Los Blancos are reportedly prioritising Paul Pogba.
If Pogba were to move to Madrid, United themselves are thought to be considering a move for Eriksen.
But first on United's list is Aaron Wan Bissaka, who they reportedly told Sporting Lisbon they would try and secure before making a bid for Fernandes.
However, with Spurs stepping up their efforts for the Portuguese midfielder, United will likely double down and think about making an offer.
Sporting's valuation stands at £70 million, which is thought to be optimistic - with £45 million the more realistic figure.
Fernandes scored 31 goals from midfield last season.
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