That is a common question these days as the risk of blood clots with the J and J vaccine is being compared to the risk of clots with certain birth control methods.
All combination contraceptives ( a combination of estrogen and progesterone, with estrogen being the more risky for clots ) have a small risk of blood clots. These aren’t the clots you see during your period, but the ones that form and block blood flow in important blood vessels- deep vein thromboses in legs (DVT), clots in lungs that can be fatal (pulmonary emboli or PE) or blockages in the vessels of the head and neck , which lead to strokes.
Anyone on the pill, patch or ring needs to understand that these clots are a rare, but dangerous complication of estrogen ( also occur during pregnancy and in women using estrogen replacement therapy for menopausal symptoms) and to know the warning signs. If you are on these medications and have a hot, swollen, tender leg without a recent injury- you need to seek medical care immediately to rule out a clot, because a clot in the leg can travel to the lungs or other organs and cause major issues.
That said, these methods of birth control are still very safe for most people and pregnancy carries a higher risk of clot than these medications. For women wanting to avoid these risks, or women with certain medical issues that make them more prone to blood clots, there are a number of progesterone only methods of preventing pregnancy- progesterone only pills, IUDs, implants and more.