The Rams have secured a "record" deal with their shirt sponsor, online casino 32Red, "off the back of" the former England captain joining the club.

Rooney was then given squad number 32, but the company denies it part of the sponsorship arrangement.

"Clubs need to be very conscious of that link with gambling," Adams said.

"It did raise an eyebrow when I saw the 32 number on Wayne Rooney's shirt.

"Hopefully Derby County will be mindful and will be talking to the FA because we have to make sure we look after those who are vulnerable.

"There is problem gambling in our country so we have to be very careful about the message that football is sending."

Football finance expert Dan Plumley last week said the deal added to the "ethical issue" football has with the gambling industry.

Half of Premier League clubs have primary shirt sponsors that are gambling companies, which is the joint highest seen in England's top flight.

The rate of such deals are higher in the second division, as Derby are one of 16 Championship clubs whose main sponsor is a betting firm, 15 of which are displayed on the front of their shirts. 32Red themselves are on the front of four shirts - Derby, Leeds United, Middlesbrough and Preston North End.

Last week a gambling expert said a "loophole" in regulations protecting children from being targeted by betting companies was being exploited.

Although 32Red will not appear on any replica children's shirts, as prohibited by regulations, they can have Rooney's name and number 32 printed on the back of their kit.

The betting company has said their agreement "complies with FA regulations".

'Betting is part of football'

Adams said that football clubs need to "be mindful about the impact of problem gambling on the people that go to football matches and people that buy the shirts, youngsters in particular".

He went on to say that the government have taken "some action" on the gambling industry by cutting the maximum stake on fixed-odds betting terminals to £2 earlier this year.

"We'll be keeping a close eye on it, but at this moment it's an issue for the football authorities to deal with," said Adams.

The Premier League's acting chief executive Richard Masters told BBC sports editor Dan Roan that 'betting is part of the game' and sees no reason for football to act.

"Our football clubs have had relationships with betting companies for a long time," he said.

"There are guardrails with how those relationships can be activated and our clubs adhere to those very carefully.

"We also have betting rules in place and lots of integrity measures to protect the competition. We ourselves, the Premier League, do not have a relationship with a betting company.

"I don't think we're at a position yet where we're concerned we have to intervene and nor does the government. Betting is part of what brings people into football, it's only a small part of it but it is part of the game."

The Football Association last week declined to comment as Rooney is yet to come under their jurisdiction, because England and Manchester United's all-time leading goalscorer does not move back to the English game from Major League Soccer side DC United until January.

BBC Sport have contacted Derby County, but is yet to receive a response.