Mount stepped off the bench in the 67th minute against Bulgaria at Wembley Stadium on Saturday to make his international bow as the Three Lions secured a 4-0 win in their latest Euro 2020 qualifier.
That saw the former Derby County loanee line up alongside Rice for the first time since both were in the Chelsea academy, with the pair having played alongside each other for six years until Rice was released at the age of 14.
The duo have remained close and holidayed in Dubai together over the summer following Rice's breakout campaign at West Ham.
Mount is looking to make a similar impact across London this term, and has scored twice in his first four Premier League appearances under Lampard.
And Rice is backing the 20-year-old to continue that form in front of goal as he grows into a top-flight regular.
"I’ve said to loads of people, he could score in every game if he wanted to," Rice said. "The positions he gets himself in, I can tell he’s training with Frank and they are working really well together because he’s like a second striker, making those runs which Lampard used to.
"He’s pretty much immaculate. He’s playing the same way Lamps used to play. I am not surprised he is doing that for Chelsea and he is going to keep doing that there for years.
"Since we were young, in training and games, we have always been egging each other on to get better.
"We have always had a winning mentality. We always want to do well. We went our separate ways but always stayed in contact, dropped each other a message now and then, and it just shows the type of character he is that he has come on and done excellently, and he has started at Chelsea like that.
"It was special [to see him debut]. From the moment he was lined up to come on, I had a smile on my face. When he ran out and the reception he got, it was really special. I don’t know how he was feeling but I felt like a proud best friend. It was really special for him and his family."
On playing alongside his good friend, Rice added: "We never spoke about it until this summer. The season he had at Derby was outstanding, he did really well with the Under-21s and the call-up from Gareth [Southgate] was waiting for him. When we were on holiday together this summer, we said, ‘What if it happens?’ with a smile on our faces.
"When the squad came out the other day, we had a phone call to each other, we were really happy and we have been enjoying it in camp as well. That’s the main thing. I think this would be the last way of us playing together.
"You would maybe think we would rekindle [our partnership] at a club somewhere but for it to come with England is really special. It is the best way to do it."
Rice himself impressed as he completed the full 90 minutes on Saturday in what was just his fourth appearance since committing himself to England over the Republic of Ireland.
But, with just seven games to go before the Euros kick-off next summer, he is well aware he needs to continue to take his opportunities if he is to be selected for his first senior international tournament.
"It is not a lot of time at all. When you get the tournament, you need to be in the best frame of mind for the team. Gareth’s got decisions to make for the squad but, with seven games to go, everyone is fighting for a place and it is healthy and everyone wants to do well.
"I know I have got to get better for sure. We have had a good start to the season at West Ham. I know I can go up another level. It is only the start of the season.
"Tonight, I have probably had the most touches of the ball I have had all season, so I want to keep getting on the ball as much as I can, keep it ticking, intercepting the ball, reading the game well – pretty much my all-round game I am trying to improve.
"I am looking forward to Euro 2020 and hopefully by then I have gone up another level."