Scottish-born Arfield, who represents Canada at international level, has started 15 times for the Clarets this season but could leave Turf Moor this summer.
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The 29-year-old was linked with West Ham in January, and another two unnamed English Premier League sides but Burnley boss Sean Dyche was hopeful of the ex-Huddersfield Town star signing a new contract.
Dyche said in January: “We’ve had talks with [Scott]. He’s sat tight on an offer so we’ll keep open minds, as will he I hope, to what comes next.
“I’ve been keen for him to stay for a long time. Scott knows where it all stands, he knows the club’s situation and he’s been a very good player for us and continues to be.”
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Three months on Arfield is still to sign a new deal with the side sitting seventh in the English top flight.
And Rangers, who are understood to have been looking at the midfielder during the last transfer window, are admirers of the former Scotland Under-21 international.
However, Arfield’s wages - thought to be in the region of £20,000-a-week - could be a stumbling block for Rangers if they pursue a deal.
The Ibrox side have also been linked with Crystal Palace midfielder James McArthur, but the boyhood Gers fan - whose Eagles contract is up in the summer - could look to remain in England after the summer.
On-loan QPR midfielder Sean Goss is expected to return to his parent club in the summer while Dalcio will also return to Benfica B. The futures of Carlos Pena and Harry Forrest, on loan at Cruz Azul and AFC Wimbledon respectively, are also uncertain.
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