Canadian citizens have long enjoyed special rules for entering the United States on various types of visas. Most notably, the majority of Canadian citizens do not need to visit a U.S. consulate in advance of their travel in order to obtain a visa to enter the United States for a temporary stay. Canadians also enjoy access to preflight inspection procedures, in which they clear U.S. immigration and customs checkpoints before boarding the plane. This both saves time and hassle if they are ultimately denied entry into the United States.
When it comes to green cards, however, Canadians are in most respects treated the same as any other immigrants to the United States. But their access to certain special types of temporary visas, such as the TN, might help them establish a relationship with a U.S. employer that is willing to eventually sponsor them for a green card.
The exception is American Indians born in Canada, who are considered to have a right to enter the United States, and can apply for a green card after proving their lineage. This includes all Canadian-born persons with 50% or more American Indian blood.
Under the United States-Mexico-Canada Agreement (or originally, the NAFTA treaty), Canadians (as well as Mexicans) who practice certain professional occupations and who have a job offer from a U.S. employer may apply for TN status. It lasts for up to three years, and can be renewed in up to three-year increments, with no limit on renewals. The U.S. government keeps a list of the qualifying occupations.
While in the United States on a TN visa, it might not be acceptable to apply for a green card. Some visas, such as the H-1B allow "dual intent"—that is, simultaneously assuring the immigration authorities that you are planning a temporary stay while also pursuing the possibility of a green card —but the TN is not one of them. Your best bet is most likely asking the employer for whom you are already working to petition for you to receive an H-1B visa, and then to seek U.S. permanent residence, potentially through that employer.
If you are eligible for a green card due to your American Indian blood and birth in Canada, you must reach its Contact Center and make an appointment with a U.S. Citizenship and Immigration Services (USCIS) office within the United States.
You can enter the U.S. via a land border without a visa or passport, but should have your tribal identity card ready to show the CBP Officer. If arriving in the United States by air, you will be required to carry a passport, as well.
At your USCIS appointment, you need to will request that a record of admission for permanent residence be created for you. There is no particular application or form that you need in order to apply for a record of admission for permanent residence in these circumstances, and no fee. You also must submit the following items to USCIS:
Other non-Indian Canadians must apply for permanent residence in the United States in much in the same manner as citizens of other foreign countries. Commonly used eligibility categories include through a job offer in the U.S. or a close family tie to a U.S. citizen or permanent resident.
One advantage that Canadian citizens do have, however, is that it is easy for them to enter the United States legally. With a legal entry, it becomes much simpler for those who become eligible for a green card, for example through marriage to a U.S. citizen, to get through the various procedural hoops and apply for the green card without having to leave the United States. You'll still need to be wary of committing visa fraud, however; namely entering the U.S. with assurances to the border officers that your intention is a temporary stay, when the truth is you plan to apply for a green card. See, for example, Do Canadians Need a K-1 Visa to Enter the U.S. & Marry a U.S. Citizen?.
WARNING: As a result of the COVID-19 pandemic, Canadian and other travelers seeking to enter the United States through land ports of entry and ferry terminals at the U.S.-Mexico and U.S.-Canada borders must be fully vaccinated against COVID-19 and be ready to provide proof of vaccination if it's requested.
If you are a Canadian citizen and wish to seek permanent residence in the United States, you might wish to seek counsel from an experienced U.S. immigration lawyer. The attorney can advise you as to the easiest and most efficient path to U.S. permanent residence and help you prepare the paperwork and evidence needed.