Propose and suggest mean virtually the same thing when they are used to mean "put forward a plan or idea for [someone] to think about":*
....We have to suggest/propose a list of possible topics for next term's seminars.... Might I suggest/propose that you offer your manuscript to Collins?*
Likewise, they may both be used when recommending a person for a certain position, or a place to visit: Helen has suggested/proposed Richard as the next chairman of the society....Can you suggest/propose somewhere for a short holiday?*
Propose, however, is more formal in style than suggest.
Only suggest can be used to mean "imply":
-- Are you suggesting (not proposing) that my professional reputation has been compromised?
Only suggest can be used to mean "indicate":
-- The findings suggest (not propose) that a deficiency of these nutrients can have serious consequences in later life
Propose, not suggest is used in certain expressions:
-- To propose (not suggest) a toast to someone -- To propose (not suggest) [marriage] to someone
Propose is also used intransitively to mean "intend to do something":
-- He proposes (not suggests) to follow the Lewis and Clark trail for 500 miles
*Adapted from Collins COBUILD English Language Dictionary (Collins, 1999)
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