The £1.5m-rated 25-year-old was strongly linked with a move to Middlesbrough last week, with Jacob Butterfield rumoured to be coming the other direction in a separate deal.
But the transfer to Teesside for the former Leeds man has been put on hold.
And Brighton, who made a failed £1m bid for Clayton last January, are said to have been alerted.
Clayton was left out of the squad for the 4-0 home defeat by Bournemouth.
The club took the unusual step is issuing a pre-match statement to explain the situation.
It read: “Over the last three days, Huddersfield Town has been in advanced talks with another club about a possible permanent transfer of Adam Clayton.
“In that period Adam has been discussing personal terms with this club, meaning he has not participated in Town’s pre-match preparation.
“As a result, he has been left out of today’s matchday squad.
“A deal remains possible, but has not yet been concluded.”
Former boss Mark Robins had Jonathan Hogg, Oliver Norwood and loan man Radi Majewski as his starting midfield on Saturday, opting to leave £375,000 signing from Liverpool, Conor Coady, on the bench.
Town remain in talks with free agent Republic of Ireland centre-back Sean St Ledger over a possible deal.