Spurs offered Spanish side Celta Vigo close to £9m for the 20-year-old midfielder but will need to bid more if they want to lure the highly-rated youngster away from his Spanish club.
Lyon have started with a larger bid and offered a more lucrative deal personally for Pape Cheikh but it is understood that he is more open to moving to London.
Tottenham fans are still awaiting their first summer signing this year but the north London side are closing in on a number of targets with the deadline fast-approaching.
Ajax defender Davinson Sanchez, who is rated at more than £30m, is one player they are in talks with as well as lining up a £3m deal for Southampton back-up goalkeeper Paulo Gazzaniga.
They have also been linked with Paris Saint-Germain defender Serge Aurier but are one of a number of Premier League teams linked with the Ivorian international.
Pape Cheikh has a £25m release clause in a contract running until 2021 but both Spurs and Lyon are set on securing the player's signature for significantly less.
The midfielder was born in Senegal but has represented Spain at Under 17 and Under 19 levels after moving there aged 14.