Robinson, 18, has already spent time on trial at Finch Farm and now Everton are keen to tie up a deal for the youngster who is already been labelled the 'new Seamus Coleman'.
Everton paid just £60,000 to Sligo Rovers back in 2009 to acquire Coleman, who has gone on to establish himself as first choice at Goodison Park and one of the best full-backs in the Premier League.
Right-back Robinson could now be ready to follow Coleman across the Irish Sea.
"Ryan certainly has a big future in the game," said Shelbourne boss Johnny McDonnell.
"Everton would have seen a rawness to him but he has a lot going for him.
"We'd have no problem allowing him to move on to a better level in England. Everton are chipping away with us on Ryan at the moment and we hope to get the deal over the line soon.
"The initial fee won't make us millionaires but we'll make sure it's heavy on the end of it. It might be better for us to go for the add-ons and everything else."