The flying winger may struggle to link up with his hometown club this summer.
Anya enjoyed a superb season last year as he helped the Hornets into the Premier League for the first time in eight years. The 27-year-old featured in 35 Championship games, setting up seven goals, as Watford just missed out on the title, but went up as runners-up.
Celtic have been chasing a winger to join Ronny Deila's squad in recent weeks, and in Glasgow-born Anya, they had been hopeful they had the perfect player.
The wideman has never played professional football in his home city, but has been involved with the Scotland side for the last couple of years, and scored a goal against World Champions Germany in last September's narrow 2-1 loss in Dortmund.
Record Sport report that the Bhoys had begun negotiations with representatives of both Anya and Sunderland striker Steven Fletcher recently, with the hope of getting them in before the Champions League qualifiers, but that Watford's asking price of £4 million is beyond what they are willing to pay.
With Anya about to enter the final year of his contract, the Record states that chief executive Peter Lawwell is looking to pay around half that amount for a player of Anya's age, while Watford are refusing to budge on the former Celta Vigo man's valuation.
Unless Celtic front up for the player, or manage to persuade the Premier League new boys to lower their asking price, it looks like the Scottish Premiership champions will have to look elsewhere.