Welsh has slipped down the defensive pecking order at Parkhead and it has been thought that Celtic are prepared to let him move on this month.
Dundee United got in touch with Celtic to try to take Welsh to Tannadice on a loan agreement for the rest of the season.
Celtic though rejected the proposal from Dundee United and it appears Welsh will be staying at Parkhead until at least the summer.
Dundee United will now look elsewhere for a late defensive reinforcement with the window open until midnight in Scotland.
Welsh has also attracted interest from south of the border, with Sheffield Wednesday keen.
The 23-year-old has made just six appearances for Celtic over the course of the season so far, being booked twice.
Welsh clocked the full 90 minutes in a Champions League group stage defeat away at RB Leipzig in October.