The subject of the sentence is "The President", which is singular. The phrase "along with the Council of Ministers" is a modifying phrase and does not affect the subject-verb agreement. Therefore, the verb should also be singular.
"wishes" is the singular form of the verb and agrees with the singular subject "The President".
• "wish" is plural and would be incorrect here.
• "will wish" changes the tense unnecessarily.
• "is wishing" is awkward and not appropriate in this context.
Hence, the correct form is: "The President, along with the Council of Ministers, wishes to visit India next week."