The Toffees supposedly sent a €25m (£21.5m) proposal to the Italian side for the 25-year-old earlier this week, which was swiftly rejected.
Sean Dyche's side are refusing to give up on the Portuguese attacker, though, and according to Fabrizio Romano, Everton have now come forward with a revised offer.
Romano states that the Toffees are now willing to go as high as €30m (£25.7m), although Udinese have instead pointed towards the €35m (£30m) release clause in Beto's contract.
However, negotiations are still progressing between the two clubs, who are now said to be working on the add-ons structure for a striker who has netted 22 goals in 65 matches for Udinese.
Everton went down 1-0 to Wolverhampton Wanderers on Saturday afternoon, marking the first time ever that they have started a Premier League season with three successive defeats to nil.
Over at the City Ground, Nottingham Forest are apparently close to sealing an £8m deal for Chelsea outcast Callum Hudson-Odoi.
The former Bayer Leverkusen loanee — who failed to reignite his career in Germany last term — has been exiled from Mauricio Pochettino's squad and will leave before the window slams shut.
Everton, Fulham, Lazio and Forest have all been tipped to move for Hudson-Odoi — as have teams in Saudi Arabia — but according to football.london, Steve Cooper's side are on the verge of winning the race.
The 22-year-old is set to join the Tricky Trees on a permanent basis after Fulham baulked at Chelsea's £8m asking price, but he is yet to agree personal terms with Forest.
However, a transfer is expected to go through in the coming days, as Hudson-Odoi prepares to depart Stamford Bridge following a
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