You will need an Assumable mortgage.
Assumption is a process in which one or more new borrowers are added to an exisiting mortgage in place of one or more old borrowers. The lender must approve of the substitution... which will be just like the process of application and approval of the original borrowers.
FHA loans are assumable. Other mortgages can be made assumable if they include a "Novation" clause. "Novation" is a legal term meaning substitution of parties.