Majority of the scholars agree that the days need to be made up before the next Ramadan. The evidence is the action of A’isha (ra) who made sure she did it before the next Ramadan.
Abu Salamah narrated that A’isha (ra), said, “I had to make up some fasts I missed during the month of Ramadan, but I was unable to do so until the month of Sha’ban as I was busy with the Prophet (pbuh).” [Bukhari and Muslim]
The concept of performing a missed act of worship before starting a new act applies to all forms of worship. For example, if a person misses duhr prayer and then the time of asr comes in, he needs to pray duhr first and then asr.