What is the synonym for the word propose?

This one really depends on the context.

'Propose' as in marriage: fire the question, get down on one knee, pop the question, ask for (his/her) hand in marriage.

'Propose' as in suggest: Suggest, put forward, advise, proposition, recommend, urge.

There are also other contexts that 'propose' can be used in, these other contexts have synonyms of their own.