Mr Johnson, a former media analyst at Kleinwort Benson and a columnist in Sunday Business, will invest up to pounds 250,000 immediately, with a view to taking complete control by the middle of next year with a total investment approaching pounds 1m.
The deal is thought to have been brokered by the paper's editor, Anil Bhoyrul, and comes in the wake of tension over the future of the title between Mr Bhoyrul and Gordon Brown, leader of Group 2000, a consortium of Leeds-based investors that owns the title.
Sunday Business went into administration in June, owing pounds 4m. Group 2000 bought the company from the administrators for pounds 50,000 in September, after lending it pounds 1.3m. Relations between Mr Brown and senior staff have deteriorated, recently leading to the resignation of Mr Rubython as editor.
Mr Johnson said reports of a deal were "premature". "We are talking, but I don't want to reveal my hand," he said. Mr Brown was not available.
Mr Johnson and his partner Hugh Osmond specialise in investing in struggling companies, reviving their fortunes and floating them. Their most successful investment has been Pizza Express, which they took control of in 1992 when it was making profits of pounds 2.2m. Last year, it made pounds 6.6m, and in April Johnson cashed in options worth pounds 1.15m in the company.
Tomorrow night, Mr Johnson will host Sunday Business's Christmas party - at a West End branch of Pizza Express.