Howe has been linked with the Toffees vacancy - although he began the week far down the list of contenders at 25/1.
The 40-year-old was subsequently cut to 8/1 by Thursday, behind Marco Silva and Andre Villas-Boas, and has since climbed above the latter to 6/1.
Everton sacked Sam Allardyce earlier this month and are widely reported to have made Silva (who is still the front-runner at 1/4) their number one target.
However, it is possible that the Merseyside club could turn alternatives, amid speculation that Silva's former employers Watford could try to complicate proceedings.
Childhood Everton fan Howe has taken Bournemouth from League Two to the Premier League - where a fourth consecutive season awaits - across two spells as boss.
Villas-Boas is now 7/1 to be Everton's next manager, ahead of former Toffees midfielder Mikel Arteta 8/1.
However, no other candidate currently occupies odds shorter than 20/1.
Could Everton turn to Bournemouth's Eddie Howe as their next manager?