I first read about this connection in the mid-90s in a post in the Classic RMIM by Dr Gurbakshish Singh Bagga (=guri). Sahir was inspired by Amir Khusro to write a few lines in that famous Roshan Qawwali, Ye Ishq Ishq hai Ishq Ishq.
Original from the PhD thesis on Amir Khusro by Sayma Hasan Khan, Hindi Department, Aligarh Muslim University, 2008.
I will transcribe the last eight lines of this poem called, Varsha Ka Agman (=Arrival of Rain) by Amir Khusro in praise of Hazrat Nizamuddin Chishti.
Bahut Kathin Hai Dagar Panghat Ki
Kaise Main Bhar Laaoon Main Madhwa Se Matki
More Achche Nizam Piya - Kaise Main Bhar Laaoon Main Madhwa Se Matki
Zara Bolo Nizam Piya
Paniya Bharan Ko Main Jo Gayi Thi
Daud Jhapat Mori Mataki - Pataki. Bahut Kathin Hai ...
Khusro Nizam Ke Bali - Bali Jaiye
Laj Rakho More Ghunghat Pat Ki
Now You Know!