The Scotsman performed brilliantly for his nation against Poland and Croatia in their outings this past week, with the winger tormenting Manchester City and Croatia defender Josko Gvardiol during their sides' clash last Friday.
Scotland would go on to win their match against Croatia 1-0 thanks to John McGinn's 86th-minute goal, but it was Doak that managed to capture headlines following his breathtaking performance.
The right-winger has also impressed during his loan spell at Middlesbrough so far this season, but Lynch believes that barring a major uptick in terms of direct goal contributions, Doak is likely to leave the Reds in the near future, telling Sports Mole: "It was my understanding there was an openness to selling him in the summer just gone. That was something that was on the table.
"I feel like it would take something huge for that [openness to selling him] to change this season and he'd have to do more in this loan spell at Middlesbrough - score more goals, set some more goals up, and those numbers aren't quite there.
"The fact that [the club] did make that call last summer, even though it didn't end in a permanent exit, I just think the writing is on the wall for him at Liverpool."
Limited playing time
Doak's pace and trickery on the right has led some to suggest that he could be the long-term successor to Mohamed Salah, who is set to leave the club for free at the end of the current campaign as things stand.
However, there is still a chance that the Egyptian will extend his stay on Merseyside, and that could be problematic for the 19-year-old's development, an issue compounded by the fact the Reds already boast a number of options in the forward line.
When asked about whether Doak would be given the chance to establish himself in the first team under Slot, Lynch told Sports Mole: "I'm not sure that he's a guaranteed future star at Liverpool. I think they're just too well stocked in those forward positions now - they've got six there, they probably can't offer him the minutes.
"The big question mark is, if you are going to be Liverpool level, it is [about] end product. You look at Mo Salah's output, that is what Liverpool need on that right wing.
"I don't think it's one where he's going to come back and be a first teamer, not least because if [Slot] signs Salah to a new contract, he's got Chiesa in front of him, he's got Salah in front of him."
Given Salah's commitment to conditioning and fitness, as well as that Federico Chiesa is expected to fill in for Liverpool's top scorer at times, it is difficult to see how Doak would get the minutes required to flourish in a situation where three players would be jostling for the same position in the starting XI.
Still room for development
So far in the Championship, Doak has netted once and registered three assists in 10 appearances for Middlesbrough, and while that haul is fairly modest, it is the winger's dribbling ability that has caught the eye.
In fact, only Real Madrid's Vinicius Junior and Paris Saint-Germain's Bradley Barcola have carried the ball into the opposition penalty area more often than the Scotsman.
That fact was not lost on Lynch, who insisted that the forward still has lots of room for growth as a footballer despite the journalist's scepticism regarding Doak's future at Anfield, saying: "He's an unbelievable dribbler. Really good off both feet, so he can really go either way and that puts confusion in defenders' minds.
"I'm not writing him off by any means. He's a very, very young lad and there's a lot of development to come.
"He still can have a really, really good career and be one of those who has a slightly delayed climb to the very top of the game, and maybe Liverpool will regret it in the long term."
Doak will hope to continue his strong start to the season and if he can add more end product to his game during his loan, then the teenager could get the chance to work his way into Slot's plans at Liverpool.