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McCullum was unveiled as England's new Test coach at Lord's on Friday, speaking at length about his hopes for reinvigorating the set-up. Although the squad named for next week's first Test against New Zealand featured a number of conservative selections, McCullum hinted that he was keen to talk to several players who have had greater success for England in the limited-overs formats.
Buttler endured a miserable Ashes tour to Australia as the Test wicketkeeper, but has rediscovered his form at the IPL, where is currently the leading run-scorer in the competition. Moeen, who retired from Test cricket last year, also enjoyed a productive IPL, as did the uncapped Livingstone; Rashid has not played a Test since the 2018-19 tour of the Caribbean, citing a desire to stay fresh for white-ball cricket.
"Jos is one of those players you look at instantly and think how could he be so dominant in one form of the game and not quite have found his feet, other than a few fleeting performances, in Test cricket?" McCullum said.
"There's certainly guys you look at and think there's a lot of talent that could improve the side given the right opportunity. There's no reason why, if you're good at T20, you can't bring those skills into Test cricket and you look at some of the guys who have dominated the IPL in the last two months. It's just a matter of trying to identify how they're going to do that."
On the question of whether Moeen would be open to a Test return, McCullum said: "We'll see. I'm sure if Mo wanted to, and was prepared to put the yards in to make it back into the side, then he'd challenge, no doubt." The Daily Mail subsequently reported that Moeen had spoken to McCullum and was ready to make a comeback in whites.
"I'm not sure someone like Adil would play every game every year anyway," McCullum added. "But my mentality is, if they're the best cricketers, why not have the conversation and see where you get to?"
Livingstone was previously picked for the Test squad on the 2018 tour of New Zealand, but has since made his name on the global T20 circuit. He scored England's fastest T20I hundred, from 42 balls, against Pakistan last summer and was a member of the team that got to the semi-finals of the T20 World Cup. At this year's IPL, he had the highest strike rate of anyone to score 400 runs.
"Again, a conversation needs to be had about what his appetite for Test cricket is," McCullum said. "If he is keen on playing, then how does he see himself getting into the side?
"Just because you're a good cricketer, it doesn't mean you automatically get selected. You've still got to earn the right to be able to do so. I don't have any pre-conceived thoughts about whether he's good enough or not good enough. I just look at him and think he's a very good cricketer, and think it's worth having a conversation.
"Obviously you've got the likes of Livingstone, Moeen, Rashid. All these guys have played international cricket before, been successful in the other forms of the game and you'd think they'd be able to transition across.
"I look at them and think there'll be a time where they may get an opportunity if they're invested enough. There's probably been a bit of hurt at times for those guys too, because they've been in the team at times and then out of the team at others and there hasn't been that persistence for them."