Relations between institutional investors and activists have evolved in recent years, and it is not unheard-of for big investors to support activists who have set their sights on a high-profile company. In 2013, T. Rowe Price teamed up with Mr. Icahn to oppose a leveraged buyout of Dell.

Southeastern Asset Management, another institutional investor, worked with Mr. Icahn on another offer to buy Dell, and also supported an effort by Barington Capital Group, a hedge fund, to shake up the retailer Dillard’s. And ValueAct Capital, a hedge fund that focuses on technology investments, took a stake in Microsoft knowing it had the support of other big shareholders, ultimately winning a seat on the board.

Neuberger Berman has taken a more confrontational approach before. Last year it campaigned for change at another company in which it had invested, the technology company Ultratech. It secured two board seats, but the process, Mr. Amato said, was “in all honesty expensive and time-consuming.”

In the case of Whole Foods, Neuberger Berman lobbied for the activists to do the messy work of battling the company in public.

At the same time, activists, and Jana in particular, have grown more comfortable working with institutional investors. Before Jana went public in 2012 with its fight against Agrium, the Canadian fertilizer company, it met with other shareholders and sought their support. And the year before, when Jana was pressuring the publisher McGraw-Hill to spin off part of its business, it did so with the support of the Ontario Teachers’ Pension Plan.

But it is less common for an institutional investor to share its work on a specific target with activists in the way Neuberger Berman did with Whole Foods. Mr. Ackman has described receiving occasional calls from institutional investors hoping to pique his interest in a company. There is even a term for the interplay: “R.F.A.s” or “requests for activism.”