Kosiniak-Kamysz said Hegseth had told him that discussions on the issue were "going in the right direction."
The Polish minister said he interpreted the response as a "green light" from the Pentagon for a permanent US military base in Poland.
He added that the final decision rested with the White House and was expected in early December.
Kosiniak-Kamysz, who is also a deputy prime minister, described the visit as potentially the most important and a breakthrough in the history of recent visits to the United States by Polish defence officials.
He said Poland had become not only "a model US ally" but also a "strategic partner."
The talks also covered the possible purchase and use by Poland of US-made Tomahawk cruise missiles and their launch systems.
Kosiniak-Kamysz said Poland had submitted a request several months ago and had received a positive response from the Pentagon.
The final approval rests with Congress, he said.
The BGM-109 Tomahawk is a cruise missile used by the US Navy and Britain's Royal Navy. It can strike targets more than 1,000 kilometres away, Polish state news agency PAP reported.
Kosiniak-Kamysz also said the two countries had signed an agreement on cooperation between their defence industries, including development projects and joint initiatives.
He said Poland wanted to help ensure that US equipment produced by Polish-American companies could qualify for European Union funding. He added he would "present a proposal on the issue" at an upcoming meeting of EU defence ministers.
Asked about efforts to build a new European security architecture, Kosiniak-Kamysz said Poland was no longer discussing such a framework but actively building it.
He described his Washington visit as an important part of strengthening security across Europe, particularly NATO's eastern flank.
"I cannot imagine giving in to Russian pressure in this area," he said.
Kosiniak-Kamysz's visit came a week after US midterm elections in which President Donald Trump's Republicans lost control of both chambers of Congress.
Kosiniak-Kamysz said he hoped Democrats would help play a positive role in strengthening NATO's eastern flank and supporting Ukraine.
He said he saw no opponents of a permanent US base in Poland among either Republicans or Democrats and added that the Polish government maintained contacts with both sides of the US political spectrum.
(gs)
Source: IAR/PAP