Ronald Koeman was dismissed by the Toffees hierarchy on Monday after the Merseyside giants slipped into the relegation zone following a heavy 5-2 loss to Arsenal.
Paddy Power value 46-year-old Dyche as second favourite to assume the role left by Koeman, while caretaker boss David Unsworth remains favourite as he prepares his side to face Leicester City at the King Power Stadium on Sunday.
The Newcastle United legend believes Dyche is the obvious choice, as well as criticising Koeman's decision not to replace the loss of Romelu Lukaku.
"The single biggest reason for their problems this season was that they didn’t replace Lukaku, who had everything they’re lacking," Shearer said. "They had to try and get somebody in who could give them a focal point, and score the all important goals.
"I would love Everton to give a British manager a chance, Sean Dyche for example.
"He has done an absolutely magnificent job at Burnley but he won’t stay there forever and might think that he’s taken them as far as he can."
Everton will be without midfielder Idrissa Gueye for Sunday's clash against Leicester, while long-term absentees Seamus Coleman, Yannick Bolasie and Ross Barkley remain sidelined.
Three points and only their third win of the season could see Everton climb out of the relegation zone, with the Toffees joining Stoke City, West Ham United and Swansea City on eight points.