Proxicom, which lists AOL Time Warner, General Electric and Merrill Lynch as clients, cancelled the merger agreement it made with Compaq after it received a counter bid from Dimension Data earlier this week.
Dimension Data's offer at $7.50 per share was 30% higher than the Compaq bid.
Karen Cramer, head of investor relations at Dimension Data, said that the premium price was aimed at deterring Compaq and at persuading Proxicom's board to consider a second bid.
"Our offer is designed to discourage Compaq from getting into a bidding war with us," she said. "We also had to offer a price that would remain attractive to Proxicom."
Compaq has until Monday to match Dimension Data's offer or pull out. If Compaq walks away it will pick up a $10m (£7m) termination fee from Proxicom.
Compaq's profits have fallen with the decline in the PC market and the firm is looking to expand into high margin areas like IT consultancy and services. Acquiring Proxicom could help it enlarge its global services portfolio.
Dimension Data, which is based in South Africa and listed in London, operates in more than 30 countries. Acquiring Proxicom would give it the foothold it needs to make an impact in the US.
Sandy Fitzpatrick, a senior analyst with Canalsys, said: "Both companies are after Proxicom's services and client base. Its services and skills complement both.
"Integration with Compaq would be a different experience from joining with Dimension Data. They are very different organisations. It is not just a commercial issue for Proxicom, which is probably looking at where its people would best survive."
Anthony Miller, an analyst with Ovum, said: "In terms of Compaq, this is a bit like Hewlett-Packard trying to acquire PricewaterhouseCoopers. Whether this is going to be an appropriate fit for them, I don't know. Compaq services have not performed well under Compaq ownership in the past and Proxicom may find there is a culture clash.
"Compaq traditionally offers lower level services and Dimension Data is more medium- to higher level. Proxicom would find integrating with Compaq like fitting a square peg in a round hole, whereas Dimension Data has a services culture."