Sporting, Wolverhampton Wanderers and Southampton have each seen offers knocked back in recent days, with the packages said to run into eight figures whether structured as an outright sale or a loan carrying an obligation to buy. Spurs have so far shown no willingness to negotiate on any of them.
The 21-year-old's stock has risen sharply on the back of his year in the Championship, where he scored 12 goals despite Oxford being relegated, collecting the club's men's golden boot and Young Player of the Year prizes in the process. Across 47 appearances he was involved in 16 goals in total, adding to the impression he made in his breakthrough season at Spurs, when he scored in the Europa League and made three senior Premier League appearances.
Now training at Hotspur Way having added physical bulk during his season in the Championship, Lankshear is set to be assessed by Tottenham manager Roberto De Zerbi on the training pitches this week, with a decision on his next step possible before Spurs fly out on their summer tour to New Zealand and Australia.
De Zerbi is expected to want a new senior forward this summer, with Richarlison entering the final year of his contract, and any incoming addition could have a knock-on effect on Lankshear's game time given the club's more limited European fixture list this season.
The striker's situation could, in turn, shape the club's plans for 18-year-old Mason Melia, with the Republic of Ireland international's own loan prospects potentially linked to what happens with Lankshear.
Spurs are understood to want a significant fee before considering a sale of the homegrown forward, with his value expected to continue rising. Attention now turns to whether the interested clubs return with improved offers, and what De Zerbi makes of the 21-year-old once he sees him training alongside the first-team squad.