Negredo has scored 70 times in 139 appearances for Sevilla across four seasons, and featured for Spain in their Confederations Cup campaign.
Spurs have been linked with a move for the Spanish forward for some time, but it is Manchester City who appear close to securing his services. Atletico Madrid were also believed to be interested in the 27-year-old.
According to reports Manchester City chief Ferran Soriano has met Sevilla president Jose Maria Del Nido to discuss a deal, which may be completed soon.
However The Mirror claim Spurs are once again showing a 'strong interest' in landing the hot-shot striker as they look to bolster their front line for the new season.
Tottenham suffered a lack of firepower last season and are expected to land a striker this summer.
So far Barcelona's David Villa is said to be the man for the job, but with Fiorentina also interested the north London club may have to look elsewhere.
Whether they turn to Negredo remains to be seen, and reports suggest that they may have left it to late to beat Manchester City, who are said to be confident of getting the deal over the line.