The 26-year-old, who spent last season on loan at the Kassam Stadium, has one year left on his current Everton deal, but looks likely to join the Trotters.
Garbutt didn’t look convincing in the first half of the season for Oxford, but saying that, neither did the majority of the squad.
A return of injured players, an upturn in form and shifting Garbutt from a left-back to a left-winger were all pivotal in the U’s strong finish to the campaign.
It was this turn around in form that raised Robinson’s eagerness to re-sign the former England Under 21 player.
But, according to the latest report, Garbutt looks set to sign for Paul Lambert, who looks to build a squad ready for an immediate return to the Championship.
The versatile left-sided player has made over 150 football league appearances in various loan spells away from Merseyside at Wigan, Colchester, Fulham and Oxford.
It will be another bitter blow for Oxford, who also lost out on the acquisition of former loanee, Jerome Sinclair.
The U’s are looking to bolster their squad. Alex Gorrin and Tariqe Fosu are the only senior additions, whilst Robinson’s side have been linked with Chris Cadden, Stuart Findlay and Ched Evans among others.