Rovers made the approach on Thursday, with the Cherries accepting the undisclosed fee on Friday morning.
Talks have opened with the 28-year-old, but a deal is not expected to be concluded quickly.
"We understand that this will be a big decision for James at this time in his career and acknowledge the fact that other clubs will be interested in his services," read a statement from Doncaster.
"The deal is a long way from being finalised and further negotiations will take place once James returns from a short break with his family."