the lord of the rings - Did the people of Minas Tirith know that Isildur had a living heir?

When Aragorn came to Minas Tirith, he was very quickly accepted as the King Returned. Part of this was the prophecy "The hands of the king are the hands of a healer, and so shall the rightful king be known". But this legend seemed to only be remembered by one nurse, and then spread through the city.

‘King! Did you hear that? What did I say? The hands of a healer, I said.’ And soon the word had gone out from the House that the king was indeed come among them, and after war he brought healing; and the news ran through the City.

It seems kind of surprising that Gondor would accept Aragorn so quickly, especially given their love of (the new Steward) Faramir and also the clearer links of Prince Imrahil to the city.

Were the people of Minas Tirith aware that Isildur had living heirs among the Dunedain? Did they literally expect a King to eventually return, or was it a hopeful legend, more like the hopes of the Ents that the Entwives would eventually return?

Answer

No, it was basically just legend at that point. From Appendix A:

Each new Steward indeed took office with the oath 'to hold rod and rule in the name of the king, until he shall return'. But these soon became words of ritual little heeded, for the Stewards exercised all the power of the kings. Yet many in Gondor still believed that a king would indeed return in some time to come; and some remembered the ancient line of the North, which it was rumoured still lived on in the shadows.

(Return of the King Appendix A: "Annals of the Kings and Rulers")

Gandalf also indicates this in a conversation with Denethor (emphasis mine):

[T]he rule of Gondor, my lord, is mine and no other man's, unless the king should come again.'

'Unless the king should come again?' said Gandalf.'‘Well, my lord Steward, it is your task to keep some kingdom still against that event, which few now look to see. In that task you shall have all the aid that you are pleased to ask for.

(Return of the King Book V Chapter 1: "Minas Tirith")

However, there were probably only three Gondorians who knew for sure, and they all learned about it within a year of Aragorn ascending to the throne:

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  Boromir, who learned about it at the Council of Elrond:

  
  

  
  

  'And who are you, and what have you to do with Minas Tirith?' asked Boromir, looking in wonder at the lean face of the Ranger and his weather-stained cloak.

  
  

  'He is Aragorn son of Arathorn,' said Elrond; 'and he is descended through many fathers from Isildur Elendil's son of Minas Ithil.'

  
  

  (Lord of the Rings, Book II, Chapter 2, "The Council of Elrond")

  
  

  
  



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  Faramir, who learned it from Frodo:

  
  

  
  

  'If any could claim [the Ring], it would be Aragorn son of Arathorn, whom I named, the leader of our Company from Moria to Rauros.'

  
  

  'Why so, and not Boromir, prince of the City that the sons of Elendil founded?'

  
  

  'Because Aragorn is descended in direct lineage, father to father, from Isildur Elendil's son himself. And the sword that he bears was Elendil's sword.'

  
  

  (Book IV, Chapter 5, "The Window on the West")

  
  

  
  


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  Denethor, who presumably learned about it through the palantír, or deduced it from Pippin's mentions in their conversation:

  
  

  
  

  'Do I not know that you commanded this halfling here to keep silence? That you brought him hither to be a spy within my very chamber? And yet in our speech together I have learned the names and purpose of all thy companions. So! With the left hand thou wouldst use me for a little while as a shield against Mordor, and with the right bring up this Ranger of the North to supplant me.'

  
  

  (Book V, Chapter 7, "The Pyre of Denethor")