The company also reported better-than-expected third-quarter results and saw its price target upgraded at a number of Wall Street firms.
"We're well aware that possible combination or alternatives could accelerate shareholder value," Chief Executive Michael Pizzi said on a conference call.
"If that's the case, we remain open really to all discussions as we always have been around the direction that we should take to drive that value."
E-Trade posted earnings of $1.08 a share compared with the FactSet-derived consensus analyst estimate of $1.01. Third-quarter revenue of $767 million compared with the estimate of $743 million.
UBS lifted its price target on E-Trade shares to $47 from $41. Analysts at KBW lifted their price target to $40.50 a share from $37. Jefferies raised its price target to $46 from $44 and reiterated a buy rating.
Meanwhile, JMP analysts reiterated a market-perform rating and $50 price target.
E-Trade shares were rising 4.5% to $40.80 in the premarket.