Tay's value is about £24m but it has become vulnerable to a takeover as a result of falling profit and its low share price.
Country & Metropolitan last week announced it had bought 1.5 million shares in the Yorkshire-based firm, taking its stake to 4.13 million shares, or 15%.
Tay told its shareholders this week that as there was no "sensible, fundable" offer from Country & Metropolitan, their interests would be best served by sticking with Tay.
But a City source claimed Country & Metropolitan was poised to swoop for Tay's remaining shares.
He said: "Country & Metropolitan is looking to acquire more shares. It wouldn't make sense to come this far and only take another 10%. A takeover offer is on the cards."