The Football Association's medical department spoke to Spurs on Tuesday and are awaiting updates on Kane, who is suspected to have suffered ankle ligament damage.
Spurs snapped up Llorente from Swansea City back on deadline day in August 2016, with Mauricio Pochettino swooping to land the Spaniard after his 15-goal campaign last season.
Spurs have a league clash against Newcastle to reschedule, while their trip to Brighton on Friday April 20 would also get postponed if they reach the FA Cup's last four.More news: Betting Preview For 2018 NCAA Tournament — March Madness Odds
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Given the delay for a diagnosis, it looks highly unlikely that the 35-goal striker will be fit for Spurs' FA Cup quarter-final at Swansea on Saturday, while his involvement in next week's global break also looks doubtful with England set to face Holland and Italy in friendlies. They also go up against table toppers Manchester City at Wembley Stadium on April 14.
Gareth Southgate is set to name his squad for this month's global friendly matches against Holland and Italy, but Kane will be absent for both ahead of this summer's showpiece.
In March 2017, he hurt his ankle during an FA Cup win over Millwall and was ruled out for four weeks.
The Tottenham striker was taken off in the first-half of their fixture against Bournemouth on Sunday with damaged ankle ligaments - the third he has suffered the same injury in 18 months.