His current deal is due to expire in the summer and as a result the 30-year-old will now be on the lookout for another club in time for the new season.
"Very disappointed to say I won't be signing a new contact at Shrewsbury Town. I would like to thank the club and fans for a great 2 years," he wrote on Twitter.
"I wish you all the best next season and I will always be looking out for your results. Thank you for the great memories."
Morgan joined Shrewsbury in 2011 and played a key role in their promotion to League One last year.