Boro, armed with cash to spend following the big-money sales of Djed Spence and Marcus Tavernier, to Spurs and Bournemouth respectively, are in the market for several players ahead of the closure of the window on September 1.
Boro made enquiries about the 21-year-old earlier in the summer, only to be snubbed. An offer has now been turned down by Hull, who want to keep the player, who has been in discussions over a new contract.
Greaves’ current deal runs until next summer, although the club does have the option to extend it by a further 12 months.
While being keen to retain the services of Greaves, it is understood that a bid of £7m-plus could entice the club to potentially do business.
Speaking last week, Boro manager Chris Wilder spoke of his desire to bring in five new signings – with the Teessiders chasing at least two new forwards.
Bringing in a left-sided centre-half is another priority for Boro, with Greaves firmly on their radar. Watford are also interested in the centre-half.
Meanwhile, Rotherham United manager Paul Warne is expecting a spate of transfer activity at Championship clubs ahead of the summer deadline – specifically in terms of loan signings.
The Millers chief wants to bring in several more players before September and is especially keen to bolster his strikeforce.
Summer signing Tom Eaves is currently sidelined along with young forward Josh Kayode.
Warne, whose side visit Coventry City on Sunday, said: “Permanent deals have been harder to come by.
“I’ve met loads of players and unfortunately they’ve chosen to go somewhere else.
“I don’t think we’re in a bad place but, as always, I’d like to have a bit more done.
“The loans are happening late, as is always the case. There will be loads of those done in the last three weeks of the window.
“The Premier League clubs have kept their young players until they’ve been on tour.
“Once they come back, they start filtering their players out to Championship clubs, then Championship clubs start filtering out players to League One side et cetera.
“We’re not fully at the loan trough yet. We definitely need a couple of loan signings.”