But club bosses believe an increased offer will be enough to force Lyon's hand, claim the Mail.
Spurs are seeking forward reinforcements as manager Mauricio Pochettino plots an overhaul of his front line, with both Emmanuel Adebayor and Roberto Soldado expected to be sold.
And while they negotiate a deal with Lyon over N'Jie's future, Spurs are also reportedly interested in a Premier League swoop for two other forwards.
The Mexico international, 27, is free to leave Old Trafford at the right price after spending last season on loan at Real Madrid.
Pochettino also appears to be hedging his bets on landing a striker as he steps up his pursuit of West Brom's Saido Berahino.
The Standard also suggest a bid for Berahino is imminent, although they may have to offer upwards of £20m to bring the Baggies to the table.
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Spurs hope a £10m bid for Clinton N'Jie will force Lyon to sell their star
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Lloris, 28, has been tipped to replace United goalkeeper David De Gea if the Spaniard forces through a move to Real Madrid.
Spurs are determined to retain their France international but Schmeichel insisted United should step up their pursuit.
"If De Gea goes I would love Lloris to come to Manchester United. He has exactly the style of play and he is proven in the Premier League," he told talkSPORT.