Constable, United’s top scorer for the past six seasons, was offered a new deal, while the club activated a clause in Wright’s contract which ties him up for another 12 months.
The other major outcome from meetings at the Kassam Stadium yesterday saw Deane Smalley, Sean Rigg and Scott Davies released.
Michael Raynes’s future had already been decided. Although the centre back’s initial contract expired this summer, his 31 appearances this season were enough to activate an extra year.
Tom Newey was the one player who did not have a meeting yesterday, with the left back due to speak to Waddock today.
It was the first step to Waddock moulding the squad for next season, with Constable and Wright the highest profile players whose contracts expired this summer.
Waddock is keen to see the experienced pair, who have made more than 470 appearances for the U’s between them, remain for next season.
He said: “Both Jake Wright and James Constable have been tremendous servants to this club and want to enjoy success here.
“They have an opportunity to be part of the rebuilding and the exciting times that I believe lie ahead for Oxford United.”
The moves were warmly welcomed by most supporters, particularly concerning the popular Constable, who is two goals away from becoming United’s all-time top scorer.
But it was the end of the line for three players.
Smalley was United’s second highest scorer this term, with nine, but a return of 16 goals in three injury-hit seasons fell short of expectations.
Rigg barely missed a game in his first season after signing from Port Vale in 2012, but only showed glimpses of his best form in this campaign.
The emergence of Callum O’Dowda in the latter weeks of the season also contributed to making the left winger surplus to requirements.
Davies was only a fringe player under Chris Wilder, but his chances appeared to be boosted by the appointment of Waddock, for whom he had played at Aldershot and Wycombe Wanderers.
But although the midfielder featured more prominently in the final weeks of the season, it was not enough.
Waddock said: “It’s never an easy decision and we wish the lads who we have let go well in the future.
“At the same time, we draw a line under last season, learn valuable lessons, and look to build a side capable of challenging next time.”
The final decision surrounded Wayne Brown, who had operated under a player/goalkeeping coach role this season.
The 37-year-old, who has been at United since 2011, will not be offered playing terms, but discussions will take place about a coaching role.