Sweett was fined £2.25mln in February for the bribery offence, which took place in 2010, and has since stopped operating in the Middle East.
WSP Global said the deal would beef up its presence in the UK while also bringing in project and cost management services expertise.
“The combination of skills and compatible cultures will provide a great foundation for the two organisations to work together constructively,” said Paul Dollin, WSP’s chief operating officer.
The offer is 35p per share in cash, a 52% premium to the share price at the close yesterday.
Douglas McCormick, Sweett’s chief executive and who joined last year, added: “By joining WSP, this will provide the company with a stronger platform, both operationally and financially, for growth in the years ahead."
Shares in Sweett rose 49% to 34.3p.