Left-back Shaw made his senior international debut in the 1-0 win over Denmark on Wednesday, replacing Ashley Cole at half-time, but it was Lallana who attracted praise for his performance.
The midfielder came off the bench to replace Jack Wilshere on the hour mark and set up the only goal for Daniel Sturridge with a chipped cross to the back post.
Shaw feels his team-mate is a certain pick for a place in Roy Hodgson's final 23-man squad for the tournament.
"I think Adam is nailed on for Brazil with the way he's been performing for his club and now his country," he told Southampton's official website.
"We know what talent Adam has and he proved it to the world (against Denmark).
"It's a good kind of attention. Every kid would dream of getting this recognition from media all around the world.
"It's great for me but I'm just going to stay myself. I'm not going to change after playing for England, so I'll just keep doing what I’m doing."
