Arfield, whose contract at Turf Moor is up in the summer, has been linked with Rangers and is expected to sign a pre-contract with the Ibrox side by the end of the week.
Speaking ahead of Burnley’s match with Brighton on Saturday, Dyche said: “Rangers asked about Scotty Arfield, what his position was.
“Scotty was a boyhood Rangers fan. That one’s pretty simple in its form, if they want to speak to him. We said, ‘Yes, of course’. We understand it.
“He’s done a lot for this football club and we hope things work out in his favour. We want him to have a bright future and I think there’s a good chance of the move happening.”
Paying tribute to Arfield and his fellow midfielder Dean Marney, who is also set to leave the club, Dyche added: “We have to allow these lads the right, and the chance, to align their futures and honour the fact they’ve been fantastic servants to us.
“They have been a massive part of what we’ve done here and fantastic servants of Burnley Football Club.”
Arfield left Falkirk for Huddersfield in 2010, joining the Clarets in 2013 and twice helping them win promotion to the Premier League.
He has scored 22 goals in 194 appearances for Burnley.