The Reds lost 2-0 at Stamford Bridge in the fifth round of the competition, with Willian and Ross Barkley providing the goals.
While Robertson and Gilmour were on the opposite side in the capital last month, the pair may be on the same side in the future for the Scottish national team.
Robertson is the captain of his country, while Gilmour has yet to pick up his first senior cap – and the Chelsea kid revealed what the red backs told him after the final whistle at Stamford Bridge.
“After the Liverpool game, I swapped shirts with Andy Robertson,” Gilmour told Sky Sports.
“He came up to me after the game and said,” Well done, you played really well, keep doing it and we hope to see you in the first team soon. “
“It was great to hear that from the captain of the country. So I asked him if we could exchange shirts and he said “no problem, let’s do it indoors”, so it made me buzz. “
Gilmour added: “Everyone knows how I feel about representing my country.
“I hope I’m not far away. I just need to keep working hard and see where it takes me. “