The linked rings on every Chinese Coke bottle and the leaping athletes on each McDonald's paper bag testify to the power the world's biggest corporations believe this summer's Olympics wields.

But having spent huge sums, the companies sponsoring the Beijing games are about to find themselves the targets of a new, more vigorous war on China's human rights record by campaigners boosted by the success of protests along the torch relay route.

Yesterday a coalition of Tibetan groups warned Coca-Cola that it would be "complicit in a humanitarian disaster" unless it used its influence to ensure Tibet was dropped from the torch route. And tomorrow, Dream for Darfur will launch a critical "report card" on sponsors of the games.

Campaigners are urging companies to press the International Olympic Committee and Beijing itself for change - or risk damaging their brands. "Companies [who do not act] will get physical protests; they will get letters; we will ask people to turn off their adverts," said Ellen Freudenheim, director of corporate outreach at Dream for Darfur, which argues that they should press China to put pressure on Sudan as its major oil buyer.

"Sponsors don't make policy and we understand that. But combined they have about the equivalent of the GDP of Canada, the world's eighth largest economy; they have government affairs offices; they have lobbying firms; they have international presences - and they all do engage in politics.".

Canny activists are targeting the stars who represent the brands too: George Clooney has already said he has raised the issue of Darfur with Omega, the Olympic sponsor and watch manufacturer which he advertises. The aim is to create a domino effect as spokespeople or consumers pressure sponsors, who in turn push the IOC into lobbying China.

Each of the 12 global partners for this year's event have paid £30-40m for a four-year deal. An IOC spokeswoman said yesterday: "A number of companies engage in partnerships with the IOC and the Olympic Movement to help us fund the work we do and spread the Olympic values. Their support is key not only to the success of the Olympic Games but also to the sustainability of the Olympic Movement.

In the period 2001-2004, sponsors contributed $1,459m, 39% of the IOC's revenue.

Activists believe their protests are already having an effect. The angry reception afforded the Beijing torch relay in London, Paris and San Francisco earlier this month caused acute discomfort to the relay sponsors Coca-Cola, Lenovo and Samsung. Last week, Human Rights Watch accused "cowardly" partners of "remaining "largely silent" in the face of abuses; just a few days earlier the media freedom body Reporters Without Borders disrupted Coca-Cola's annual general meeting.

Campaigners say some sponsors are raising concerns privately.

"Realistically, everyone who signed up for Beijing knew there were various risks involved," said Damien Ryan, a Hong Kong-based media consultant advising several sponsors. He acknowledged that this "risk factor has escalated". Activists are well aware that multinationals hope sponsoring the Beijing games will give them privileged access to 1.3 billion increasingly wealthy people without entrenched purchasing habits.

"Almost all of the top level sponsors want to leverage the games to a better market position in China," said David Wolf, president of Beijing-based corporate advisers Wolf Group Asia.

Olympic sponsors argue it is simply unfair to hold them responsible for every action by the Chinese authorities.

The sportswear giant Adidas, a Beijing rather than IOC partner, said in a statment it was "conscious of the exceptional importance of the protection of human rights". "Sponsors, however, should not be expected to solve political issues. We clearly see the limits of our influence."

A Coca-Cola spokesperson pointed out that the soft drinks giant had supported the Olympics since 1928, expressed "deep concern" for the situation in Tibet and cited its support for charities working in Sudan.

Amnesty has asked all Beijing games partners to raise human rights concerns directly with the IOC and Beijing.

"The universal declaration on human rights calls on every individual and organ of society, which includes corporations, to ensure human rights are respected. Corporations do have influence, and we would call on them to exert it publicly," said Robert Gooden, Amnesty's Asian-Pacific campaign co-ordinator.