Campbell has confirmed holding talks with Celtic but Arsenal are understood to be lining up a one-year contract extension for the 35-year-old.
Brown accepts appointing a new manager is the most pressing concern for the Hoops, but he would be pleased to see a big-name player arriving in the east end of Glasgow.
"I think it's important to get a manager in place before we sign anyone but I'm sure there will be a few big names coming in as well," Brown said.
"Yes, his experience would really help.
"He has played in the Champions League final and won Premier Leagues with Arsenal so it would help us at the back.
"We can attract big people, we are a big club, so the bigger the better for us."
Brown was more circumspect when discussing the possibility of Rangers' out-of-contract striker Kris Boyd making the move across the city - a story which has been categorically denied by Celtic.
Brown pointed to Rangers striker Kenny Miller, who played for Celtic before returning to Ibrox for his second spell at the Govan club, and said: "I heard about it but I don't really know what's happening to be fair and I'm sure no-one will be coming in until we get a new manager.
"But it happens sometimes. Kenny Miller has done it so you never know what could happen.
"I'm sure Kenny found it hard at the start but he has done really well and he did really well at Celtic.
"He has been a great player and wherever you go, you try to give 100 per cent."
Brown was more strident when asked about the need for Celtic to get a manager in place for when the players report back for pre-season training on July 1.
The former Hibernian midfielder insists the players "hope" that Neil Lennon, who finished his period as interim boss on a high with eight successive SPL wins, is given the post on a permanent basis.
Asked if he thought the Irishman should get the job, Brown replied: "I think we all hope so, because of the job he has done in the last eight games.
"It's the best form we've hit in two years so he deserves a chance.
"He has done everything he possibly can and it shows that we have fighting spirit left in us.
"I'm sure the club will have it sorted out by July 1 so the club can go flying into pre-season.
"It's up to the club but it would help us, we would know where we stood.
"But we are quite relaxed and quite open about it and whatever the board decides we will be happy."
Brown was also relaxed with the fact that Walter Smith, who captured his second successive SPL title for Rangers last season, has signed a new one-year deal.
Brown said: "It's not up to us what happens (at Rangers).
"We have to go out and play our own game and hopefully we can have a better season this year than we did last year.
"I think by the end of the season the team spirit was better. We brought in a lot of players and it was hard for us when they first came in.
"But we have started to gel. We don't want to be known as unlucky all the time, getting beaten in finals or not making it to finals.
"We are here to win as many trophies as we possibly can and that's what we look forward to doing."
Celtic remain keen to buy Landry N'Guemo from French club Nancy.
They club have until Monday to exercise their option to make the midfielder's loan deal permanent.
From June 1, N'Guemo, heading for the World Cup finals with Cameroon, will be back on the market, where the French club believe they can find a club willing to pay their valuation of £1.8million.
Celtic believe that, having already paid a substantial loan fee for N'Guemo, they might get the player for £1.2million.