The firm said on Tuesday that it was seeking talks with Galliford Try bosses to come up with a deal.
Galliford Try rejected Rok's bid of 51p a share, understood to have been made late last week. Analysts described it as "opportunistic" and said 60p was more realistic. One said 70p would be a fair price for the group.
A Galliford Try statement said: "The board does not believe that this proposal could be recommended as it significantly undervalues Galliford Try."
It is understood that Rok, which bought regional contractor Llewellyn last year, is chiefly interested in the firm's construction arm. It would sell on Galliford Try's profitable housebuilding division to reduce debt.
City analysts called on the Galliford Try group to separate its construction and housebuilding businesses at the start of the year.
Rok's statement to the stock exchange said: "Rok is seeking to commence a constructive dialogue with the board of Galliford Try with a view to agreeing terms and beginning formal due diligence as quickly as possible."
The firm is aiming to reach the £1bn turnover mark by 2007 and its strong track record in turning around struggling businesses puts it in a strong position to acquire Galliford Try.
After it bought Llewellyn for £16.25m, Rok cut 150 jobs. When it announced its results in August, the firm said the acquisition had contributed to a doubling of its pre-tax profit to £3.4m for the six months to 30 June.
However, sources at Galliford Try have argued that the fortunes of the group's construction division is already improving. It is being run by managing director Andy Sturgess, who joined from Skanska at the beginning of the year.
The construction arm made a pre-tax profit of £1.6m for the six months to 30 June.
Sturgess, who pledged to increase its margins to 2% within three years, said last month: "The message is that we have turned a corner."
Rok's bid led to a 6.5p jump in Galliford Try shares to 45.75p and a 10p drop in it own to 246p.