The present invention concerns an isothermal reactor (1) comprising a pressure vessel (2) closed at the opposite ends by respective bottoms (3, 4), a reaction zone (9) in said pressure vessel (2) in which at least one catalytic basket (10) is positioned, and at least one heat exchange unit (13) embedded in said at least one catalytic basket (10), each heat exchange unit (13) comprising a plurality of heat exchangers (14) each having an inner chamber intended to be crossed by an operating heat exchange fluid, the reactor being characterized in that it comprises means (17, 18) for picking up samples of operating heat exchange fluid from groups (16) of pre-established exchangers in each heat exchange unit (13), so as to ascertain the possible existence of damaged exchangers (14) in said groups (16) of exchangers through analysis of respective samples of operating heat exchange fluid. The invention also concerns a method for detecting the existence of damaged heat exchangers in an isothermal reactor of the aforementioned type.