Stavanger-based Spike Exploration has made an offer equivalent to 162 pence per Bridge share.
That saw shares in Bridge close up 34.5p at 156.5p yesterday, giving the company a market capitalisation of around £99.3m.
The cash offer is a premium of 41% compared to the closing price on September 13, which was the last trading day before the offer was announced.
Bridge said shareholders in control of around 62% of its issued share capital have already accepted the offer with 34% of those giving irrevocable undertakings.
An offer document is expected to be sent to shareholders by the end of the month.
Tom Reynolds, Bridge Energy's Aberdeen-based chief executive, admitted the company had considered making its own acquisitions or raising further funds, but the board feels the agreement with Spike offers the best return for shareholders.
He said: "This is the result of quite a lot of meetings that have been going on throughout the summer and we looked at a pretty broad church of strategic options.
"We think it is a fair deal for the business which is why we have wholeheartedly recommended it.
"By tendering their pre-acceptances several of our largest shareholders have already recognised that this is the optimal path to realising value without further operational risk and additional funding."
Mr Reynolds said he believes Bridge has great potential but "it would be two to three years to see that value really develop" into returns for shareholders which is why the cash offer is attractive.
If the deal goes ahead, Spike has agreed with its private equity backer HitecVision - which has previously indicated it will invest around £190m into Spike's growth plans - to carve out the combined UK business as a separate entity.
While Mr Reynolds is uncertain of his own future if the transaction concludes he does believe the UK staff at Bridge, with 15 people based in Aberdeen, can look forward to an exciting future.
He said: "The Norwegian part of our business will get rolled up in Spike while the UK will become a new business and I would expect much of the team from Bridge UK will be involved in that.
"Spike and their backers are looking to invest over a long period of time and it gives all parts of our business funds to grow."
Bridge, set up in 2010 and with a dual Oslo and AIM stock market listing, has four producing assets giving around 1100 barrels of oil equivalent per day although that is expected to grow to 10,000 within five years.
It is an operator on the Victoria field while also holding minority stakes in the Cormorant East, Boa and Duart fields.
A drilling programme in the final quarter of last year found three commercial oil discoveries, meaning it has made 10 discoveries from the 18 wells it has drilled.
Spike, which also has an office in London, has five Norwegian exploration licences plus three in the UK including a 15% stake in the Athena field in the outer Moray Firth area of the UK Continental Shelf.
Analyst Sam Wahab, from Cantor, believes the agreed tie-up between Bridge and Spike is a further signal the UK and Norwegian regions of the North Sea are ripe for consolidation given the "abundance of smaller players, large portfolios, and under-capitalised work programmes".
Mr Wahab highlighted other "undervalued" North Sea operators such as Xcite Energy, Faroe Petroleum, Trap Oil and Ithaca Energy.
He said: "Given the current global political climate, operating in fiscally secure regions is becoming a pre-requisite for both companies and investors.
"We therefore feel that interest in the North Sea, which offers complementary terms to operators (UK incentivises production, Norway incentivises exploration), will continue to flourish."
Mr Reynolds added: "You can see the raw materials are in place for [more] consolidation across the industry.
"If you look at our peer group there are a number of companies that could acquire others or be acquired."