Lemar is one of Europe's most wanted talents after an impressive season last year which saw him score 14 times and provide 14 assists.
Spurs have not signed an attacking player all summer and rumoured target Ross Barkley is out injured for the next three months.
Monaco have resisted approaches for Lemar all summer including from Tottenham's rivals Arsenal.
The Ligue 1 side are busy bringing in their own targets including Stevan Jovetic, and could be tempted into a late sale at the right price.
Spurs have broken their own transfer record once this summer to sign Davinson Sanchez from Ajax. To sign Lemar they will have to do so again.