The Sthala Purana (Temple History) of the Sri Rangam Temple in Tamil Nadu states that the divine Vigraha (idol) of Lord Ranganatha originated by itself (Svayamubhu) and was originally worshipped by Lord Brahma in Satyaloka.
It was then handed over to King Ikshvaku on Bhuloka, for the Tapasya he performed. Since then, the divine idol of Lord Ranganatha was worshipped by the Surya Vamsa (The Sun Dynasty, which was originated by Ikshvaku), up until Rama. It is said that Rama worshipped this Vigraha, and after he came back from exile, gifted it to Vibhishana. Vibhishana on his way to Lanka kept the statue on the ground, as he had to perform his Sandhyavandana, and here is where it stays until today.
In this picture, we can see Lord Rama worshipping Lord Ranganatha.
Here, Lord Rama gifts Lord Ranganatha to Vibhishana
The story goes onto how the statue was lost in a huge flood and was retrieved by the King Kili Chola, but that is not necessary for this question.
But my question is, are there any scriptures that specifically mention Lord Rama worshipping the idol of Lord Ranganatha? If so, please cite them in your answer.