Swansea City are said to want the Celtic midfielder as a replacement for the Everton-bound Gylfi Sigurdsson.
Armstrong's contract expires at the end of the campaign, and he is said to be stalling over the length of Celtic's offer of an extension.
Stuart Armstrong of Celtic (right)
It is claimed the Scotland international could be open to a switch to the Premier League, with Swansea the latest side to be linked from south of the border.
But according to The Irish Sun, Paul Clement's side are the only one to offer cash so far, despite Southampton taking a look at him in recent weeks.
And Swansea chairman Huw Jenkins is said to be pushing both Celtic and the player for an answer.
Armstrong moved to Celtic Park from Dundee United two and a half years ago, and enjoyed a breakout campaign for the Bhoys last season - scoring 17 goals en route to a domestic treble and earning his first international caps.
Stuart Armstrong of Scotland