The phrase "astonishing claim" refers to a statement or assertion that is beyond belief or very surprising. It suggests that the claim being made is astonishingly extraordinary or unlikely. Full definition
To the skeptical eye of anthropologists, religious story, religious ritual and religious theory all make astonishing claims about the ultimate importance of a human life. (religion-online.org)
Real Madrid have reportedly «closed» the transfers of Tottenham striker Harry Kane and Chelsea forward Eden Hazard for next season, according to astonishing claims by Don Balon. (caughtoffside.com)