Kristi Noem and Corey Lewandowski Spotted Spending Time Together in Minnesota

Kristi Noem and Corey Lewandowski appear a lot closer than they might want people to believe ... because the two were seen spending some quality time together in Minnesota.

In new footage -- obtained by TMZ -- the duo is seen biking along the shores of Lake Minnetonka Tuesday afternoon before settling in at an outdoor table at The Grocer's Table, a cozy market and café in Wayzata.

Corey's keeping it casual in a tight T-shirt and a fresh pair of shades, while Kristi tried to keep a low profile in a flannel shirt and floppy hat.

Kristi and Corey later pedaled over to CōV, a local restaurant ... where they were clearly enjoying each other's company -- laughing and leaning toward one another.

Corey appears to notice a bystander filming them ... and quickly turns his head away from the camera.

An eyewitness tells us the pair was all smiles and kept close throughout the outing.

Rumors have swirled for years that Noem and Lewandowski are romantically involved -- claims both of them have denied.

In March, Noem was pressed in a House Judiciary Committee hearing about whether she had sexual relations with Lewandowski while serving as the head of the Department of Homeland Security ... a question she dismissed as "tabloid garbage."

Lewandowski served as an unpaid adviser to Noem at DHS and is no longer with the department.