Yes, both ways. It's an unusual and old-fashioned word. In the U.S. of us most of us know it from the Bible and think of it as meaning a particularly solemn or sacred promise. It is used in a technical way in U.S. law, e.g., "the landlord covenants to keep the apartment building in good repair." Use "promise" for an ordinary promise, "contract" for a legal promise, "pledge" or "vow" for a solemn promise, unless you have a good reason for using specific word "covenant." I made a promise to my grandson. I promised I would cook macaroni and cheese for him. I made a contract with my client. I contracted to complete a piece of software. I made a vow to my wife. I vowed I would love and cherish her, forsaking all others, till death to us part. When I was in school, every day we said the Pledge of Allegiance. I pledged allegiance "to the flag of the United States of America, and to the republic for which it stands." I can't of ANY time I'd have said that I made a "covenant," or "covenanted" anything.

It's possible, but it's not so common to use "covenant" as a verb. Some of the examples I've found even use "covenanted" as an adjective (ie. the past participle). There's nothing to stop you using the word either way, but it's best to check with a native speaker to make sure the sentence sounds natural. To me, "covenant" would have a very specific use.