Praise be to Allah.
We put this question to Shaykh Muhammad ibn Saalih al-‘Uhaymeen, may Allaah preserve him, who answered as follows:
It is not permissible for him to give her a massage or for her to give him a massage. Something is bound to happen, especially if they are both young.
Islam came to close all the doors that may lead to temptation and provocation for one or both parties. Massage involves uncovering part of the body and direct touching, so it poses a serious danger of temptation and the provocation of desires. You have to look for male masseurs who can do this for you when you need it it, or let some of your family members learn how to do massage so that they can do that for you. May Allaah help us and you to do that which He loves and which pleases Him.