Jefferies could offer Strachan the best bottle of Beaujolais from Rugby Park's hotel but with Kilmarnock watching every penny, the Celtic manager must pay his own way. If the visitors win to keep up their pursuit of Hearts at the top, the drinks will be on Strachan anyway.
The pair go back a long way. "I have known Gordon since he was a kid playing at Dundee," says Jefferies, recalling his own days in the mid-1970s as a defender at Hearts while Strachan was arriving on the scene. They also crossed swords at managerial level, as two Scots exiled in the English Premiership.
Jefferies quit as Hearts manager in December 2000 to take over at Bradford. His first opponent was Strachan, in charge at struggling Coventry. "Gordon's team stayed in the hotel where I was temporarily based and we had a glass of wine together the night before the game. My side went out and nicked a 2-1 win but it made no difference. We were both relegated."
Both men were subsequently sacked. Jefferies returned north in February 2002 to take over at Kilmarnock, while Strachan has taken longer to come home. The Celtic manager ended a 21-year stay in England when he agreed to replace Martin O'Neill. Jefferies has already noticed the impact made by his counterpart.
"There is no doubt that Celtic are passing it about a bit," said Jefferies this week. "That's the way Gordon works. Under Martin [O'Neill], Celtic were probably a bit more direct.
"People are happier because they are better to watch but whether they are more successful, only time will tell.
"That's what you get judged on when you are at a club that size. Gordon was a very good midfielder. He didn't want to see the ball getting knocked from back to front because he'd watch it just go over his head.
"He played in an Aberdeen side that played good football and went to Manchester United where you have to play football."
Jefferies' own team, though, is a contrast to his own robust career. Kilmarnock have been playing some expansive and fluent football this season. "We have only lost to Rangers, Hearts and Hibernian in our 10 games this term, so this promises to be an attractive game," he said.