Gloucestershire Police said the man, from Cheltenham, was arrested under the Malicious Communications Act on Wednesday night.
He was subsequently also arrested on suspicion of a racially aggravated public order offence of intentionally causing harassment, alarm or distress, a force spokesperson said.
A joint investigation between Gloucestershire Police and Northamptonshire Police was launched following allegations of racist tweets towards Adebayo Akinfenwa and Clarke Carlisle, who is also chairman of the Professional Footballers Association.
The suspect remains in police custody in Cheltenham pending further inquiries, the force spokesman said.
A Cobblers spokesman said: "We became aware of some tweets that had been sent to our players on Wednesday evening and we immediately notified the police.
"As this is a live investigation it would be inappropriate for us to comment further."
- Crime & Justice
- Society & Culture
- Gloucestershire Police
- Adebayo Akinfenwa