Griffiths, plagued by hamstring and thigh problems, has started just three matches for Celtic in 2018 and missed Scotland’s summer trip to Peru and Mexico as he underwent tidy-up hernia surgery.
The 27-year-old made his first start of the season for the Scottish champions in their 1-0 league defeat by Hearts at Tynecastle on Saturday and is in the squad for the crucial Champions League qualifier against AEK Athens in Greece tonight.
But Griffiths, who was Scotland’s top scorer with four goals in the unsuccessful 2018 World Cup qualifying campaign, has slipped down the strikers’ pecking order at Celtic behind French duo Moussa Dembele and Odsonne Edouard.
McLeish is anxious to see the 27-year-old playing regular club football again ahead of Scotland starting their Uefa Nations League campaign next month.
“It would be difficult for Leigh to keep his levels up if he’s not playing much and keeping his match fitness,” said McLeish.
“That’s a concern for us but the challenge is for Leigh to bang in the goals. It would be very difficult to pick players when they are not playing. They lose match fitness – that’s the obvious answer.
“I don’t have anything up my sleeve in terms of new strikers to come in. I wish I had. But we want Leigh back to top fitness, because when he came into the Scotland team in the last qualifying campaign he was on fire.
“Brendan Rodgers, in the beginning, had to rouse him to get to a level at Celtic. I know Gordon Strachan wasn’t particularly liking Leigh’s level until Brendan did a wee bit of work on him and then Gordon couldn’t leave him out.
“Leigh is better when he plays every week. He needs to be playing, keeping sharp and scoring goals. He had a couple of chances against Hearts at the weekend but we want him to play to the same level every week.”
If Griffiths is not considered for the Hampden fixtures against Belgium and Albania on 7 and 10 September, McLeish could again turn to Swansea City striker Oli McBurnie. The 22-year-old failed to score in his first three Scotland outings against Costa Rica, Peru and Mexico at the end of last season but will be given further opportunities.
“He could have had three goals for Scotland by now,” said McLeish. “You know how different it can be to score a goal – in the eyes of the critics and the fans it makes a difference.
“He’s done not too badly. We’ve watched him and I spoke to Swansea manager Graham Potter, who is very happy with him. But we do need someone to score goals.
“France won the World Cup with a striker [Olivier Giroud] who didn’t score a goal in the tournament. So we have to find a dynamic for goals, regardless of whether it’s the centre-forward or coming in from wide areas. We know that’s where we’re short. I can’t dig someone out of fresh air but we are keeping an eye on every centre-forward.
“As I say, Oli is playing well for Swansea, Jordan Rhodes scored for Norwich at the weekend and we’re still looking at some of the guys who played up front in Gordon’s era. But we hope Griffiths can be ready and at the moment he looks in good condition.”
McBurnie missed a penalty in Swansea’s 1-0 win against Preston North End at the Liberty Stadium on Saturday but was praised by Potter for his attitude. The Swansea boss said the striker was a “pleasure to work with”.