The 23-year-old is in contention to feature in England's first Test since their 4-1 Ashes drubbing in Australia.
Josh Tongue and the recalled Ollie Robinson appear likely to take two of the seam-bowling places, leaving Baker vying with Gus Atkinson for a spot in the team.
Atkinson has an impressive record at Lord's, with 19 wickets at just 10.94 in two previous Tests at the London ground, as well as a century against Sri Lanka, but Baker's greater speed makes him an attractive option.
An unsettled weather forecast could work in Baker's favour if there is a prospect that overs might be lost in the first of a three-Test series.
"He's quick," England coach Brendon McCullum said at Lord's on Tuesday. "I think he's going to be a cricketer that the country really gets behind.
"You can see he charges in and leaves everything out there. Some guys have a bit more of a rhythmical component to their style -- I think Sonny is full noise."
Both Baker and Atkinson were named in a 12-man squad announced by England on Tuesday, with Shoaib Bashir the lone specialist slow bowler included.
McCullum, set to give a debut to batter Emilio Gay, said Baker's bowling can touch 90 miles (145 kilometres) per hour.
"He's an exciting proposition and, if it's not in this Test, his time will come. If it is this Test, I'd expect him to do a good job," he said.
"We'll just work out whether we want that extra air speed, if all of a sudden it becomes 35 degrees and flattens out, or if we stick to what's tried and tested at Lord's and remain with the type of bowlers that will extract some movement with the overhead conditions."
Off-spinner Bashir, who has seen off competition from all-rounder Rehan Ahmed, who bowls leg-breaks, has not played for England since last year's India series.
The 22-year-old found himself surplus to requirements during the Ashes in Australia.
"I think they are both young and exciting and they've had some great days already in an England shirt," said McCullum.
"Bash has the ability to play more of a holding role in the first couple of days here at Lord's.
"As the game deepens, and if spin starts to become more prevalent, then he can go more into an attacking role. There's a little bit of tactical nuance there."
England 12-man squad for first Test against New Zealand: Ben Duckett, Emilio Gay, Jacob Bethell, Joe Root, Harry Brook, Jamie Smith, Ben Stokes (captain), Gus Atkinson, Josh Tongue, Ollie Robinson, Sonny Baker, Shoaib Bashir.
