Sheffield United have been prepared to loan out the 18-year-old Canadian this summer and several League One clubs are interested in getting their hands on him.
Sunderland laid out the red card for him earlier in the window in order to convince him to join them on loan and even Burton Albion also made a strong pitch to sign him.
But Everton have now emerged as strong contenders to sign the teenage forward and tabled a £5m bid for him last week.
The Blades rejected the offer from the Toffees and they do not want to lose him permanently this summer.
But Everton are expected to make one more effort to sign Jebbison before the window closes on Tuesday.
He is widely admired in Everton’s recruitment department and the club are keen to deliver him to Rafael Benitez.
Sheffield United lost Dominic Calvert-Lewin to Everton for a fee of £1.5m in 2016 and do not want to make the same mistake this summer.
Jebbison is even considered to be a better prospect than Calvert-Lewin was amongst the Sheffield United coaches.
