The club confirmed they had made an offer to the 47-year-old, as well as a budget, in the hopes of keeping the former Scotland international at the Tony Macaroni Arena.
Hopkin, who led Livi back to the Scottish Premiership by defeating Partick Thistle in the play-offs, has been linked with succeeding Jack Ross at St Mirren, while Shrewsbury Town have also been credited with an interest.
The ex-Chelsea and Crystal Palace midfielder has been attracting attention since leading the Lions to successive promotions, and the West Lothian club were quick to sit down with their manager at the end of the season.
A statement on Livingston’s official website said: “The club has held talks with David with a view to retaining him as manager and building on the remarkable success we have enjoyed during his time here.
“A budget and offer have been put to him today. We are very hopeful that David will see his future at Livingston.
“David has been open and honest with the Board and verified that he has had other interest. He heads off on holiday on Wednesday and will use that time to reflect and come to a decision on his future.
“The transfer window is closed until June 10th so the board is comfortable in giving David this time. We are hopeful that the offer will encourage him to remain with the club and take us forward in the Premiership.
“The club will continue to push on with pre-season schedule and our player recruitment with the help of the coaching staff at the club.”