Teymourian has been on trial with Rovers for the past week and Allardyce was keen to take a further look at the former Bolton and Fulham midfielder.
A contract offer has not yet been made, though, and so the 27-year-old has chosen to remain in the UK to pursue other options.
Blackburn are playing Rangers, AEK Athens and host club Sydney FC in the Festival of Football Tournament this month.
They then return to Britain and play a friendly away to Hearts before beginning their Premier League campaign at home to Everton on August 14.
After impressing during his week's trial at Brockhall, Iranian Teymourian has been granted the opportunity to feature in the 26-man Rovers squad that will fly out to Australia on Thursday.
Teymourian, who was released by Fulham at the end of last season following a fall-out with former boss Roy Hodgson, is desperate to secure a contract at Rovers.
The midfielder, who has also reportedly attracted interest from Championship club Sheffield United, featured in Rovers' pre-season 3-0 friendly win at Southport on Saturday as well as their 1-0 defeat to Huddersfield on Wednesday night.
And manager Allardyce, who signed Teymourian for Bolton back in 2006, is hoping that providing the 27-year-old with playing opportunities will help him rediscover his form.
"Ando has trained very well but that doesn't mean very much until he gets into game situations. It would be good if we could resurrect his career," Allardyce told the Lancashire Telegraph.
"He was very good at Southport but this (the Huddersfield game) proved to be more of a test. It was a hard test because people around him were the young players.
"It was hard to show his full talent but I think he showed enough to get him off to Australia and we will see where we go from there.
"I know what sort of player he was four years ago. I don't know why he has slipped back as far as he has but obviously his lack of football is his big problem.
"Can he overcome that lack of football and do enough in the time he is with us to secure a contract?"
Teymourian will reportedly wait until after talks with Allardyce over his future before making a decision on attending the 10-day tour.