Redknapp was at Celtic Park on Tuesday night to watch the 23-year-old midfielder make his first club start of the season and mix it with Arsenal's star-studded side inthe Champions League clash.
And despite the transfer window shutting at the end of the month, the Spurs boss isprepared to wait until the last minute before making his move with sources inEngland warning it will be short of Celtic's asking price.
Hot property: Brown
The deal could all hinge on Redknapp selling Jermaine Jenas to Aston Villa but that would leave Celtic little time for manoeuvre over any offer.
Sportsmail understands that the figure being touted for Brown has already scared off Sunderland and Bolton.
Redknapp, however, is keen to secure the combative Brown, who he sees as the missing hard man in an already talented midfield. And, while his offer is unlikely to match the record £9m fees paid for Alan Hutton and Craig Gordon, a deal could still be struck.
Former Celtic boss Gordon Strachan paid £4.4m to Hibs for Brown two years ago, a record fee between Scottish clubs, and despite being a huge admirer of the midfielder, new boss Tony Mowbray has admitted he may have to sell before buying again.
English and foreign clubs are also tracking Andreas Hinkel, Scott McDonald, Lee Naylor and Darren O'Dea, meaning the final 10 days of the transfer window could seesignificant activity at Parkhead.
Departed boss Strachan, now a Sky TV analyst, was in no doubt about how much the cash gap contributed to Arsenal's 2-0 win on Tuesday.
'It's all down to money these days,' he said. 'That's the big difference between Englishand Scottish sides. Arsene Wenger went into this encounter armed with a machine gun and Tony Mowbray had a water pistol.
'If Celtic had more money to spend on bringing in quality players, then it would be an even playing field. As it stands, there is an imbalance. Celtic's limit is probably around £4m to £5m. It's night and day, really.'
Keeping a close eye: Redknapp was at Celtic Park to see Brown in action
However, Georgios Samaras has urged the Old Firm to cling on to their Champions League dreams, despite losing to Arsenal.
A place in the group stage of the Europa League is guaranteed even if Tony Mowbray's men fail to overturn a 2-0 deficit at the Emirates next Wednesday.
Given that Celtic and Rangers have reached UEFA Cup Finals in recent seasons, there is a growing feeling that fighting for silverware in the second-tier competition is a more realistic target for our big two.
But Celtic striker Samaras last night laughed off such talk as defeatist drivel, insisting: 'I don't think that the Europa League is our level, no. That is not the case.
'Rangers and Celtic are big clubs in Europe. Both can be Champions League clubs, for sure.
'OK, so the sides have dropped out of the Champions League recently and got to UEFA Cup Finals. Those were both great runs and the teams had good results. But three years in a row Rangers or Celtic made the last 16 of the Champions League, so we have shown that we can play at that level.
'I don't see that the Europa League is a better route for the club right now. When you play for a big team like Celtic, you want to be at the top level and that is in the Champions League.'
Meanwhile, Stephen McManus made his comeback yesterday, playing 85 minutes in a 2-2 draw against Blackburn reserves.
Sidelined for three months after a minor knee operation, the Celtic centre-half could now be in the frame for the Hibs game on Sunday week.
Scotland boss George Burley will continue to monitor his fitness closely, as Gary Caldwell is banned for the September 5 World Cup qualifier with Macedonia.
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