The Leeds-born 21-year-old has netted five goals in seven games since returning to his native county from Swansea City shortly before the January transfer window closed.
Tonight’s home game with Norwich City offers a chance to add to that impressive tally as Jose Morais’s side look to boost their survival hopes with a long overdue win at Oakwell.
If he can find the net against the Canaries or Millwall, the latter being due in South Yorkshire on Saturday, McBurnie could move into pole position to get his international debut in friendlies against either Costa Rica or Hungary.
“It is a fresh start,” said McLeish, who has made six new call-ups in a squad whose most conspicuous absentees are former Leeds United and Hull City winger Robert Snodgrass and Darren Fletcher.
“There is couple of the experienced players not in the squad, but I know what they can do. I want to have a look at some other players.
“It is a couple of friendly games. It doesn’t mean we will play friendly, but I certainly want to look at players who have been doing well for their clubs.
“We want them to play the way they have been playing for their clubs and try and challenge themselves to become even better.”
McBurnie started his career at Bradford City, but moved to south Wales in a 2015 deal that brought the West Riding club an initial £250,000 fee that could double, depending on whether the striker makes 20 appearances for the Swans.
The Bantams are also due a 15 per cent sell-on fee if McBurnie leaves the Liberty Stadium, but there is no clause to cover international recognition.
He is joined in the Scotland squad by fellow new call-ups Hibernian midfielder Dylan McGeouch, Hearts goalkeeper Jon McLaughlin, Wolves left-back Barry Douglas, Aberdeen defender Scott McKenna and Manchester United midfielder Scott McTominay.
“The next step is the guys prove it,” added McLeish, who is in charge of Scotland for the second time.
“They have to now show they can live with other international teams, improve their game and see if they can raise the bar to the highest level they can possibly go.”
McBurnie has previously played for Scotland Under-19s and Under-21s.
“International football is the pinnacle,” said the former Bradford striker. “Growing up you get it drilled into you that it is the one.
“Club football is very good, but international football is the pinnacle. For your family and stuff like that, it is a really proud moment.”
As for tonight, McBurnie and Barnsley can give their survival hopes a big lift by beating Norwich.
Depending on how second-bottom Burton Albion fare at Sheffield United, the Reds could open up a five-point buffer on the relegation zone by claiming a first home win since November 4.
This is the tenth home outing since Birmingham City lost at Oakwell and head coach Morais, taking charge of his second home game tonight, believes the players are eager to put that record to bed.
“The boys are looking forward to it,” said the Portuguese. “They have deserved more than they have been getting recently and it would be good if we reach our possibilities against Norwich.
“All the games are very important, but the home games even more so. We have the support of our fans and this is an extra power for us that we can use. And we need to use that to get the three points.”
Barnsley’s run-in is largely against sides currently sitting in the middle of the table. Only Derby County of the top six are still to come, while Bolton are the only side in the bottom nine yet to face the Reds.
Otherwise, their fixture list is crammed with meetings against teams either hoping to make a late dash for the play-offs or ones marooned in mid-table.
“I don’t see this as any bigger test than other games,” added Morais. “I don’t believe they will be any easier or harder than our recent matches. The league is very competitive and we have to show our quality if we want to win games.
“We have a squad with some interesting players and I think rotation is easily understandable in certain situations. People ask if we need to win one of the next two games and I say we need to win both games. We need to get out of the situation and so we need to win games.”
Last six games: Barnsley DLWDLL Norwich City DDDDDL.
Referee: S Martin (Staffordshire).
Last time: Barnsley 2 Norwich City 1; December 10, 2016; Championship.