When converting offers and suggestions into indirect speech, we use the verbs to offer and to suggest. There is a difference between these two verbs: the person who makes an offer intends to do the action himself, and the action is an act of kindness.

A suggestion may also be an act of kindness, but not necessarily; the person who makes a suggestion may or may not intend to do the action himself.

Direct Speech Indirect Speech
He said to her, “Shall I fetch you a glass of water?” She said, “Do, please.” (“No, don’t trouble.”) He offered to fetch her a glass of water and she accepted the offer (declined the offer).
He said to her, “Suppose we go there together?” She said, “Very well.” (“No, I would rather you went alone.”) He suggested that they should go there together and she agreed (refused).

Sentences expressing advice are converted into indirect speech by means of the verb to advise.

Direct Speech Indirect Speech
He said to them, “You had better take a taxi, you may be late.” He advised them to take a taxi, as otherwise, they might be late.