British Fifa reform campaigners have called on Serious Fraud Office and other agencies to do more to tackle corruption at world football’s governing body.

Appearing before a UK parliamentary select committee inquiry into the Fifa corruption crisis, Deborah Unger of Transparency International urged the authorities to delve deeper in a bid to track down money that might have been laundered via the country.

“They could use the follow-the-money tools that are already here. Questions do have to be asked. Has any money been spent in the UK on luxury goods and services, property for example?” Unger said.

“It is very hard to go after people directly without knowing if the money has come through the UK. But you can start asking the questions.”

In the aftermath of the arrests that followed dawn raids in Zurich in May, the SFO said it would examine whether it could be useful in helping trace laundered money. But Paul Farrelly, one of the MPs on the culture, media and sport select committee, said the SFO had not asked “a single question of the FA”.

Damian Collins, a Tory MP on the committee, said a payment made by the Australian 2022 bid team to Jack Warner, former Concacaf president, was routed via the UK.

The Swiss attorney general, Michael Lauber, investigating the award of the 2018 and 2022 World Cups, has identified 121 suspicious bank transactions.

“We are also looking for means to accelerate the procedure; in this context an additional challenge might be seen in the fact some of the information of interest to our criminal proceedings is under seal,” Lauber said. “It would be helpful if parties involved would cooperate more substantially.”

The SFO has said it has “actively been assessing material in relation to this matter and has made plain it stands ready to assist continuing international criminal investigations.”

Earlier this week Domenico Scala, the head of Fifa’s audit and compliance committee, made his own reform proposals.

“The reform process needs to be top to bottom. They are circling the wagons. They don’t want that kind of reform because they don’t want the scrutiny it will bring to the past,” Unger said. “We’ve called it a crisis of corruption and we’ll stand by that. I doubt the latest reform proposals from Scala will get very far.”