https://dictionary.cambridge.org/dictionary/english/claim

I agree with Tanya - believes that, says that - are similar in meaning.

A claim is a belief or opinion expressed which has not yet been tested or proven or about which there may be some doubt. She claims to be from England but she speaks with an American accent. He claims that the train was delayed and that's why he was late for work. Claim is also used when maybe you have an accident and are seeking compensation My house was damaged by a fire and I made an insurance claim