Formal:
* Propose: implies a more formal or official suggestion.
* Recommend: implies a strong belief in the value of the idea.
* Advocate: implies a strong and active support for the idea.
* Put forward: implies presenting the idea for consideration.
* Advance: implies a more formal and developed suggestion.
Informal:
* Hint: implies a more subtle suggestion.
* Imply: implies a suggestion made indirectly.
* Mention: implies a brief and casual suggestion.
* Float: implies a tentative or exploratory suggestion.
* Throw out: implies a casual and informal suggestion.
* Drop a hint: implies a subtle and indirect suggestion.
Other options:
* Offer: implies a willingness to provide something, including an idea.
* Impart: implies sharing information or knowledge, which may include a suggestion.
* Intimate: implies a private or confidential suggestion.
* Insinuate: implies a suggestion made indirectly and often with a negative connotation.
The best synonym for "suggesting" will depend on the specific context of your sentence.