That is a possibility (as the Deuteronomistic History explicitly refers to some of its source texts), but it is unlikely because the text in Isaiah shows redactional features from the hand of the compiler of 1-2 Kings (e.g. the time phrases in Isaiah 38:1, 39:1). If one compares the two narratives side by side, there is a consistent pattern of greater detail in the text of 2 Kings than in Isaiah and there are a number of other signs that the text in Isaiah has been abridged from 2 Kings. For example:
2 Kings 20:4-11: "Isaiah had not left the middle court, before the word of Yahweh came to him, 'Go back and say to Hezekiah, the prince of my people, "Yahweh, the God of David your ancestor says this: I have heard your prayer and seen your tears. I will cure you, in three days' time you shall go up to the Temple of Yahweh. I will add fifteen years to your life. I will save you and this city from the hands of the king of Assyria, I will protext this city for my own sake and the sake of my servant David." ' 'Bring a fig poultice,' Isaiah said and they brought one, applied it to the ulcer, and the king recovered. Hezekiah said to Isaiah, 'What is the sign to tell me that Yahweh will cure me and that I shall be going up to the Temple of Yahweh in three days?' Isaiah replied, 'Here is the sign from Yahweh that he will do what he has said, would you like the shadow to go forward ten steps, or to go back ten steps?' 'It is easy for the shadow to lengthen ten steps.' The prophet Isaiah then called on Yahweh who made the shadow go back ten steps on the steps of Ahaz".
Isaiah 38:4-8, 21-22: "Then the word of Yahweh came to Isaiah, 'Go and say to Hezekiah, "Yahweh the God of David your ancestor says this: I have heard your prayer and seen your tears. I will cure you, in three days' time you shall go up to the Temple of Yahweh. I will add fifteen years to your life. I will save you from the hands of the king of Assyria, I will protect this city. This is Yahweh's sign to you that Yahweh will do what he has promised: I will make the shadow cast by the sun go back the ten steps it has gone down on the stairway of Ahaz" '. So the sunlight went back the ten steps it had gone down. [Then follows a song by Hezekiah "after his illness and recovery" in v. 9-20] Isaiah said, 'Prepare a poultice of figs and apply it to the boil, and he will recover'. Hezekiah asked, 'What will be the sign that I will go up to the Temple of Yahweh?' ".
All the highlighted material in the 2 Kings passage is missing in Isaiah, and the bit about healing Hezekiah with a fig poultice has been dislocated to v. 21-22, which is very jarring in light of the intervening prayer dated to "after his recovery" (v. 9-20) and the question in v. 22 which is answered back in v. 7-8 (note that the event itself is already mentioned in v. 8). Moreover the prayer looks forward to the recovery (v. 16), in spite of the dating.