Regular religious services have not been held there since 1934 when modern Turkey's founding president Mustafa Kemal Atatürk signed an order converting the building into a museum. But a legal challenge claims the signature was forged.
The suspect was arrested on suspicion of attempted murder.
Gunmen kill around 15 people in attack on Burkina Faso mosque
The shooter, described as a "young white man," wore body armour and a helmet, witnesses said.
The fire at the Diyanet Mosque in New Haven caused considerable damage, but there were no injuries.
At least one of the mosques was reportedly vandalised with a sledgehammer, according to a statement released by West Midlands Police.
Turkish President Tayyip Erdogan wrapped up a three-day visit to Germany on Saturday with a visit to Cologne, where a security lockdown kept supporters and protesters away from a ceremony to open the country's largest mosque.
Watch: Syrian state television shows Assad arriving at the al-Rawda mosque.
Mosques & imams accused of not fostering a "positive mindset" towards Austria, and receiving illegal foreign funding.
Chancellor Sebastian Kurz's right wing government announces crackdown on "Islamist ideology" andf foreign funding of religious groups.