Most Muslim scholars believe that it was the angels or Adam (peace be upon him) who built Masjid al-Aqsa, forty years after Masjid al-Haram was built.
In the hadeeth of Abu Dharr al-Ghafari (ra), he reported that he asked the Prophet (saw), “O Messenger of Allah, which Masjid was built first on earth”? The Prophet (saw) replied, “Al-Masjid Al-Haram”. Abu Dharr (ra) again asked, “Which was next”? The Prophet (saw) said, “Al-Masjid Al-Aqsa”. Abu Dharr (ra) further asked, “How long was the period between the building of the two Masjids”? The Prophet (saw) said, “Forty years”. [Agreed upon]
After the great flood that covered the earth in the reign of Noah, there was no trace left of Adam’s building.
Ibn Hajar al-Asqalani said in his book “Fath al-Bari” that: ((The first founder of Masjid al-Aqsa was Adam (peace be upon him), and it’s said that it’s the angels who founded it, and it’s also said Sam bin Noah (peace be upon him) and it’s said Jacob (peace be upon him) … But I have found what confirms and supports the word of those who said: Adam is the one who founded both of the Masjids [al-Haram and al-Aqsa], where it’s stated in the Book of Ibn Hisham that “after Adam had built the Kaaba, Allah ordered him to go to Beit al-Maqdis and to build the Masjid there, and so he did, and Adam’s building of Beit al-Maqdis is well-known.”)).
And there’s a Hadith Sahih narrated by Abdullah bin Amro (ra) that the Prophet (saw) said: “When Solomon, the son of David [peace be upon them] finished building Beit al-Maqdis, he asked Allah the Almighty for three things: to give him a rule that accords with God’s will, and a kingdom never to be attained by anyone after him, and that whomever comes to this Masjid to only pray in there shall step out of his sins as the day he was born.” Then the Prophet (saw) said “He was given two of these things, and I hope he has been given the third” [Narrated by Ahmad, Al-Nasa’i and Ibn Majah, and classed as saheeh by Al-Albani in the Saheeh of Awareness & Apprehension]
This hadith appears to be in contradiction to what the scholars said about Adam’s construction of Beit al-Maqdis. Yet, Ibn Hajar al-Asqalani endorsed in his book “Fath al-Bari” what Ibn al-Jawzi, al-Qurtubi, and others said that Suliman (peace be upon him) renovated and expanded Masjid al-Aqsa after it was already built since Adam (peace be upon him) just as Abraham raised the foundations of Masjid al-Haram after it was already built since Adam (peace be upon him); which was after a long time on the foundation of the two buildings and after the flood in the reign of Noah (peace be upon him) that covered the earth back then. Therefore, what is meant by this Hadith is that Suliman (peace be upon him) renovated the building and prepared it for the worshipers, and it’s not meant that he founded it, (and Allah knows best).
No other hadith of the Prophet was found that discussed the founders of the Masjid other than those two hadiths.