The Toffees are reportedly set to test Leonid Slutsky's resolve with a bid for the £7-rated 29-year-old.
Grosicki, who only joined Hull from French club Rennes in January, has rejected a move to German side Freiburg this summer and made his desire to stay in England.
Newcastle United, Watford and Crystal Palace are also rumoured to be interested in Grosicki but Everton are in pole position and continue their summer spending.
Hull sold Andrew Robertson to Liverpool on Friday, gaining midfielder Kevin Stewart in return.
Everton boss Ronald Koeman is also reportedly keen to sign Sheffield United starlet David Brooks although a bid may wait until next year.
Brooks made four first-team appearances for the Blades last year with three of those coming in the Checkatrade Trophy.
The 20-year-old, who started his career at Manchester City, helped England U20s to win the 2017 Toulon Tournament this summer, earning the Best Player award in the process.
Meanwhile, Blades boss Chris Wilder will step up his efforts to sign a new goalkeeper following a potentially serious injury to Simon Moore in Friday night's friendly at Rotherham United.
Sheffield Wednesday has downplayed talk of a £1million bid from Leicester City for star youngster George Hirst.
Hirst was left out of Wednesday's travelling party for the pre-season tour to Portugal but Carvalhal has insisted the decision was nothing to do with an imminent transfer.
Instead, the Portuguese boss said the England U18 international, who scored 40 goals for club and country last year, would have struggled to have got enough game time in competition with Jordan Rhodes, Sam Winnall, Fernando Forestieri, Steven Fletcher, Lucas Joao and Atdhe Nuhiu.
Leeds United continue to be linked with a move for Spanish striker Alvaro Negredo.
Reports in Spain suggest the 31-year-old has been told he can leave Valencia with Leeds in the market for the former Real Madrid and Manchester City man.
Negredo played on loanat Middlesbrough in the Premier League last season and could link back up with new Leeds director of football Victor Orta.
Elsewhere, Barnsley striker Kayden Jackson has joined Accrington Stanley for an undisclosed fee.
The 23-year-old forward signed for the Reds from non league Wrexham 12 months ago but failed to break into Paul Heckingbottom's side and was shipped out on loan to Grimsby Town.
Jackson struggled during his stay at the Mariners and scored just once in 20 games.