As per Mahabharata Draupadi was close with Satyabhama. Satyabhama visited Draupadi in the forest and both are mentioned as old friends. When Draupadi disguised as Sairandhri, she introduced herself as Satyabhama's former maid and later Satyabhama also visited Draupadi in Upaplavya during the peace negotiations.
Satyabhama does mention that Rukmini looks after Draupadi's sons in Dwaraka
Your sons are skilled in use of weapons. They are living happily with Abhimanyu in Dvaravati, a place they love. Like you, Subhadra loves them with all her soul. Without any conflict, she is affectionate towards them and has no worries on their account. Pradyumna’s mother (Rukmini) loves them with all her soul. Keshava instructs them, with Bhanu” and the others.
- 224, Draupadi-Satyabhama Samvada, Mahabharata.
Srimad Bhagavatam has two direct interaction between Rukmini and Draupadi
When Krishna visited Indraprastha along with his wives, Draupadi welcomed and worshipped them.
Instructed by her mother-in-law Kunti, Draupadi duly worshipped the queens of Krsna such as Rukmini, Satyabhama, Bhadra, Jambavati, Kalindi, Mitra inda. Saibya, and the pious Nagnajiti and other women-guests, with garments, wreaths of flowers, ornaments, etc. King Dharmaraja made comfortable arrangement of lodging for Sri Krsna and that of his queens, members of the army, followers and counsellors providing everyday with new objects of enjoyment.
- Chapter 71, 10th Skandha, Srimad Bhagavatam.
On another occasion, we have Rukmini narrating her marriage story to Draupadi.
Rukmini replied “Like a lion taking away its share (prey) from the herds of goats and sheep, the Lord carried me away, placing his foot on the heads (lit. covering with the dust of his feet the heads) of invincible warrior princes (like Jarasandha, allies of Shishupala) who determined to make me over (in marriage) to Shishupala, stood (ready to fight) raising up their bows (and arms). May those feet of Lord Krsna, the abode of Sri (goddess of beauty and wealth), be available to me for worship, forever.”
- Chapter 83, 10th Skandha, Srimad Bhagavatam.