Celtic are willing to sell the Scottish international for £500,000 after falling down the pecking order at Parkhead.
Livingston chairman Dominic Keane admitted he was very keen to sign the 22-year-old, but that price-tag was too much for the club.
"Stephen is a fantastic talent and I would love to have him on board at Livingston," Keane told The Daily Record.
"We contacted Celtic about the possibility of that happening but the fee of £500,000 priced the lad well out of our reach.
"I admire the boy and it is small wonder that clubs from England are queueing up to sign him.
"Realistically, English clubs are the only ones who can meet Celtic's substantial evaluation and I would fully expect to see him move south of the border.
"It is no secret we are looking to fill that left back berth and I was deadly serious about trying to bring Crainey in and continuing his football education in Scotland, even if it was only for a year.
"I know the boy may be down-hearted at having to leave Celtic but I wish him all the best wherever he goes."
Sunderland are believed to be leading the chase for Crainey south of the border.
