BES bought a 9.95 per cent in the UK mid market stockbroker for $52 million, prompting analysts to speculate that the bank may be interested in taking Evolution over.
Neither firm has confirmed the rumours, but it is known that BES has attempted to buy Evolution in the past.
BES will benefit from the acquisition as it will be able to offer its clients access to the London market.
Success in this area could mean BES is prepared to buy the whole business, which is valued at $513 million.
BES already operates in Portugal, Spain, Angola and Brazil and is aiming to expand its investment banking services.
The rumours of a takeover have been fuelled further by the fact that shares in medium-sized brokers, including Evolution, have fallen in recent months, making them more affordable.
BES' shares in Evolution could have been bought for as little as $2, analysts say.