Wendy's shares were up 38 cents, or 8.5 percent, at $4.85 in premarket trading.
A sale of Arby's would break up a $2 billion merger that Peltz orchestrated in 2008.
Arby's North American restaurants have been struggling, as the chain contends with high unemployment among the young men that make up a large share of its customers.
Sales at Arby's North American restaurants open at least 15 months fell 5.9 percent in its third fiscal quarter, following a 7.4 percent decline in the second quarter.
The company said that by shedding Arby's, which accounts for 30 percent of the company's revenues, it would concentrate on Wendy's, where sales declines have been more moderate.
Arby's has been the noose around this company's neck, said Tim Ghriskey, chief investment officer with Solaris Asset Management, which sold more than 1 million shares in the company last fall after tiring of waiting for it to sell Arby's. By splitting up this company, it's definitely one plus one equals three.
Peltz, the company's chairman and largest shareholder, said in a statement that Wendy's is the fast food company's key driver of momentum. Peltz' Trian Fund Management LP owns 18.3 percent of Wendy's/Arby's shares, according to Thomson Reuters data.
Peltz said in June that an unnamed party had expressed interest in a deal involving Wendy's/Arby's. In November, Wendy's asked Peltz to make a quick decision on the takeover inquiry so the company could resume share buybacks. It was not clear if that inquiry is related to Thursday's announcement.
The move comes at a time other fast-food companies are trying to shed assets or even sell themselves.
Earlier this week, Yum Brands Inc
In September, Burger King agreed to sell itself to investment firm 3G Capital for $3.26 billion.
Wendy's and Arby's are both trailing behind McDonald's Corp
Arby's has introduced a value menu to try to compete with Burger King and Jack in the Box
Wendy's/Arby's said UBS Investment is assisting in the sales process. The company also said it would issue preliminary fourth quarter results next Wednesday.
(Reporting by Phil Wahba in New York, Ben Klayman in Detroit and Nivedita Bhattacharjee in Bangalore; Editing by Derek Caney)