The centre-forward came off 10 minutes before half-time during his side's 2-1 defeat at Ipswich on Saturday and appeared to be holding his chest and stomach area.
Boss Steve Cotterill revealed afterwards he was taken to a local hospital, with help of Ipswich medical staff, for checks on the issue.
Cotterill said Bowman's heart rate was as high as 250bpm and that he felt heart palpitations.
On Sunday morning, Bowman said he was feeling better and was waiting on further testing.
He wrote on Twitter: "Thank you for all the messages and support they do mean a lot, feeling better this morning just waiting for further tests.
"A special thanks to the medical staff from Shrewsbury and Ipswich and also the staff at the Ipswich Hospital."
The manager said shortly after the end of the match on Saturday: "The more worrying one will be Ryan Bowman. Where he is with his health right now we don't quite now, he's at hospital now.
"He had a huge heart rate problem, he has palpitations, I don't quite know what that's about.
"When I've finished with you guys I just hope everything is OK with Ryan."
Town's defeat at Ipswich was their third in a row in League One and leaves Cotterill's team 23rd in the table after just two wins from the first 12 league matches.