Norwood left Ipswich upon expiry of his contract, with the Tractor Boys opting not to exercise the option for an additional year.
The 31-year-old, who can play all across the forward line but is most comfortable in the number nine role, scored seven goals and laid on three assists from 27 appearances in all competitions during the 2021/22 season.
The forward has held talks with the bosses of both Bristol Rovers and Oxford, with Pirates boss Joey Barton and he sharing the same agent.
With the Pirates and the Yellows in the market for a proven goalscorer, Norwood will decide where his future lies this week.
Barton guided the Pirates back to League One at the first time of asking since their relegation in the 2020/21 season.
Ipswich and Oxford finished the League One season in eleventh and eighth place, respectively, with the latter falling seven points short of a playoff place.
The Pirates lost experienced forwards Leon Clarke and Brett Pitman at the conclusion of the previous campaign, while the Yellows lost Sam Winnall, all to free transfers, and are looking to Norwood.
