But Everton still look to be in the driving seat for the Chelsea striker's signature.
It seems Chelsea manager Jose Mourinho is willing to let Lukaku leave this summer as he revamps his forward line, with Atletico Madrid striker Diego Costa reportedly his top transfer target.
Everton, where Lukaku is currently on loan, are keen to do a permanent deal for the Belgium international but it seems Tottenham have now also entered the scramble to sign the striker.
Lukaku's first-team chances will be heavily restricted under Mourinho and it seems his long-term future lies elsewhere, despite being Chelsea's top scorer this season.
The 20-year-old has hit 12 Premier League goals during his loan spell with Everton while the recognised strikers who remain at Stamford Bridge - Fernando Torres, Demba Ba and Samuel Eto'o - have scored just 25 times between them in all competitions.
And Mourinho blasted his striker's last night following Chelsea's 3-1 defeat to Paris Saint-Germain in the first leg of their Champions League quarter-final tie in the French capital.
Lukaku's exploits on Merseyside, and with West Brom last season where he netted 17 goals in 35 games, have attracted growing interest with Tottenham the latest suitors.
Lukaku's agent, Henrotay, in HNB: "I'm looking for a club. Yes, I ve talked to Levy once, but that doesnt mean Romelu will join Spurs."— Kristof Terreur (@HLNinEngeland) April 3, 2014
It is thought that Lukaku would prefer to stay at Everton, who look assured of a Europa League spot next season but still have a chance of overhauling Arsenal to finish fourth and qualify for the Champions League. Everton host the Gunners on Sunday in a crunch clash at Goodison Park.
Uncertainty surrounding the managerial situation at Tottenham next season could hamper their hopes of landing Lukaku, who according to Terreur will not have his future sorted until after the World Cup in Brazil.
And it seems Chelsea are expected to command a huge asking price for the striker, who joined the Blues from Anderlecht as a promising teenager for £18 million in 2011, despite him making only one Premier League start for the London club.