The Scotland international, 27, is currently out of the first-team picture at Championship club Coventry City.
He joined the Sky Blues in a £600,000 deal from the Black Cats in 2006. Kyle spent time on loan at Wolves last season but failed to win a permanent move after scoring just once in 13 appearances.
Pools boss Danny Wilson is keen to sign the Stranraer-born forward on loan with a view to a permanent transfer.