In mid-morning trading on the Milan bourse, shares in the bank were up 1.69 percent as dealers digested remarks by its chief executive, Carlo Messina indicating that he was keen to spend a cash pile that has risen from €13 billion at the end of last year to €16 billion.
"We are by definition a potential consolidator," Messina told the Financial Times, adding that asset managers and insurers were also potential targets, alongside private banks.
Coutts belongs to Britain's state-owned RBS, which recently announced that it wants to sell the international business of its private banking arm but has so far ruled out selling the British operation. According to the FT, Intesa is seeking to persuade RBS to part with all of Coutts.
Zurich-based Coutts International has been valued at up to $1 billion (€805 million) and has attracted firm interest from Switzerland Julius Baer and EFG. Singapore's DBS is also reportedly considering a bid.
Founded in 1692, Coutts was, for much of its history, the bank of choice for the British nobility. In recent years it has sought to secure the business of richly-rewarded figures from the worlds of sport and entertainment, regularly presenting its services at the Cannes film festival.
Would-be clients reportedly need to have at least £1 million (€1.26 million) to invest if they want to join the British monarch on the roll of customers.
RBS is 80 percent owned by the British state which rescued it during the global financial crisis with an unprecedented £45 billion bailout.
Intesa emerged from the European Central Bank's recent stress tests as one of the strongest banks in the eurozone.