Swanson said he just returned from a visit to the region, including stops in Qatar, Oman, Kuwait and Saudi Arabia, where he met with senior leaders about an array of missile defense deals and other orders.
"We used to talk about months and now they are weeks (away) that we can touch them and see them," Swanson told a conference hosted by Credit Suisse. "There is probably $4 billion or $5 billion worth of business out there that I was able to touch real quick."
Raytheon and other big weapons makers are looking to foreign orders to help offset a reduction in projected U.S. military spending over the next decade.
Raytheon's chief financial officer, Dave Wajsgras, told the conference that more than 30 percent of Raytheon's bookings in 2013 would come from international customers, with foreign orders accounting for the "high 20 percent-range" of revenues.
"Missile defense is something that is in demand in every place that I travel to," Swanson said at the conference, adding that Raytheon was talking with customers about how they could maximize their missile systems, rather than just buying components.
At the Dubai air show last month, a senior Raytheon official said the company expected to wrap up its talks about a possible Patriot missile defense system sale to Qatar "fairly quickly" and finalize the sale of additional Patriot units to Kuwait before the end of the year.
Raytheon officials have said in the past that the Qatar deal involves 11 fire units and is valued at around $2 billion. Qatar would become the fourth Gulf country to operate the Patriot system, joining the United Arab Emirates, Saudi Arabia and Kuwait.
Raytheon is also hoping to sell a ground-based air defense system for Oman.
Swanson said some of the Middle East deals the company is hoping to complete soon are with new countries, which meant margins would start off in a lower range. He did not provide details.
Wajsgras said Raytheon's overall margins, before pension contributions, would remain strong at above 12 percent over the next three to five years.
He said there was a "little bit of upward pressure" on margins given the company's strong position in foreign markets and continued cost-cutting efforts, including efforts to work closely with its key suppliers.
(Reporting by Andrea Shalal-Esa; Editing by Leslie Adler)
By Andrea Shalal-Esa