There were prophecies that spoke about the return of Isildur’s heir, but none of them really specifically spoke about Aragorn. In a lot of ways, he was the heir in the right place at the right time. The War of the Ring, coupled with his own personal motivation to reclaim the throne (and thereby earn Arwen’s hand in marriage), plus a healthy dash of good luck, made Aragorn the king that actually returned.
There are three prophecies that supported Aragorn’s claim to the throne. The first isn’t really a prophecy so much as a legend, which Ioreth points out in the Houses of Healing: “The hands of the King are the hands of a healer, and so shall the rightful king be known." Aragorn’s ability to heal the wounded form the Battle of the Pelennor Fields, especially those suffering from the Black Breath, fulfilled this cultural belief about the Heir of Isildur.
Two actual prophecies made by Malbeth the Seer also spoke about the return of a king. The first was about Arvedui, the last king of Arnor:
Arvedui you shall call him, for he will be the last in Arthedain. Though a choice will come to the Dúnedain, and if they take the one that seems less hopeful, then your son will change his name and become king of a great realm. If not, then much sorrow and many lives of men shall pass, until the Dúnedain arise and are united again.
Arvedui chose the more hopeful path (fleeing from the besieged city), and died in the process, leaving his people without a kingdom or a king. Thus the second half of Malbeth’s prophecy was set in motion - they knew that after many generations the Dunedain would arise, and an heir of Isildur would return.
The second prophecy is the most specific one, and the one that most strongly supported Aragorn’s claim to the throne. This one concerned the Paths of the Dead and the Oathbreakers that resided there. The prophecy claims that the spirits would one day be called to fulfill their broken oaths. When it comes to who exactly would call them, Malbeth said:
Whose shall the horn be? Who shall call them
from the grey twilight, the forgotten people?
The heir of him to whom the oath they swore.
From the North shall he come, need shall drive him:
he shall pass the Door to the Paths of the Dead.
As Aragorn does in fact succeed in travelling through the Paths of Dead, north to south, out of need for an army to defeat the Corsairs attacking Gondor from the south, he pretty clearly fulfills Malbeth’s requirements for the Heir of Isildur.
Whether or not Aragorn could have reclaimed the throne had the War of the Ring not happened is uncertain. It’s likely that he wouldn’t have even tried - though he was deeply in love with Arwen, and knew that Elrond would only give his blessing if he could make Arwen queen, he had up until the quest seemed rather unlikely to do anything about it. The situation was made worse, in many ways, by his time in Gondor as Thorongil - he earned the jealousy and hatred of Denethor, and feared that he could never peacefully reclaim the throne while Denethor served as steward.
SOURCES: LOTR, LOTR Appendices