Rumours have been circulating in recent months that Paulo Sousa is keen on the Irishman, and reportedly put a £500k bid in last month, prior to the end of the loan transfer deadline.
With his contract ending in 12 months time, Quinn says he's still keen to sign a new deal, but he would 'consider' a move away from Bramall Lane in the summer.
The midfielder told Sky Sports "I've been here seven or eight years now and I'd love to stay here because I am settled, with my family all around me.
"A new deal would be great here, but that depends on whether they want me and if they don't then that is when I will have to look elsewhere for a new challenge, which might be good for me.
"Swansea are a good side and it was a great compliment to be linked with them recently.
"They always play lovely football and that would suit me down to the ground, but I've got a year left on my contract here.
"I will need to speak with Sheffield United first and see if they want to renew my deal because I'd prefer to have more than a year on my contract going into next season.
"If something came up from elsewhere in the summer then I would have to consider it because I'd like to have my long-term future sorted out."