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The US Navy currently only operates THREE conventionally powered icebreakers:
USCGC Polar Star (WAGB-10) (1977-2006, 2013-; in service - Heavy Class. Able to break through ice up to 21 feet (6.4 m) thick by backing and ramming, and can steam continuously through 6 feet (1.8 m) of ice at 3 knots (5.6 km/h);

USCGC Healy (WAGB-20) (2000-; in service) - Medium class. Technologically advanced ship/only ship capable of operating in the Arctic/able to break 4.5 ft (1.4 m) of ice continuously at 3 knots (5.6 km/h; 3.5 mph) or ice 10 ft (3.0 m) thick when backing and ramming, and can operate in temperatures as low as minus 50°F;

USCGC Mackinaw (WLBB-30) (2006-; in service - Operates on the Great Lakes only/a jinxed ship to boot).

Russia operates two nuclear powered Arktika class ice breakers:
the Yamal (1992 -);
50 Let Pobedy (2007 -);

Two Taymyr class nuclear powered ice breakers:
Taymyr (1989 -)
Vaygach (1990 -);

Three LK-60Ya class nuclear powered ice breakers:
Arktika (2019 - (planned); under construction);
Sibir (2020 - (planned); under construction);
Ural (2021 - (planned); under construction)

Diesel-powered icebreakers
Project 97 (including variants):
Ivan Kruzenstern (1963-; ex-Ledokol-6
Yuriy Lisyansky (1965-; ex-Ledokol-9)
Buran (1966 -)
Semyon Dezhnev (1971-)

Ermak class:
Ermak (1974 -)
Admiral Makarov (1975 -)
Krasin (1976 -)

Kapitan M. Izmaylov class:
Kapitan M. Izmaylov (1976 -)
Kapitan Kosolapov (1976 -)

Kapitan Sorokin class
Kapitan Sorokin (1977 -; fitted with Thyssen-Waas bow in 1991)
Kapitan Nikolaev (1978 -; rebuilt in 1990)
Kapitan Dranitsyn (1980 -)
Kapitan Khlebnikov (1981 -)

Kapitan Chechkin class
Kapitan Chechkin (1977 -)
Kapitan Plakhin (1977-)
Kapitan Chadaev (1978 -)
Kapitan Krutov (1978 -)
Kapitan Bukaev (1978 -)
Kapitan Zarubin (1978 -)

Magadan class
Magadan (1982 -)
Mudyug (1982 -)
Dikson (1983 -)

Kapitan Evdokimov class
Kapitan Evdokimov (1983 -)
Kapitan Babichev (1983 -)
Kapitan Chudinov (1983 -)
Kapitan Borodkin (1983 -)
Avraamiy Zavenyagin (1984 -)
Kapitan Mecaik (1984 -)
Kapitan Deminov (1984 -)
Kapitan Moshkin (1986 -)
Karu (1988 -; ex-Karhu, ex-Kapitan Chubakov; purchased from Finland)
Tor (2000 -; purchased from Sweden)[41]
Dudinka (2006 -; ex-Apu; purchased from Finland)

Project 21900/21900M
Moskva (2008 -)
Sankt-Peterburg (2009 -)
Vladivostok (2015 -)
Murmansk (2015 -)
Novorossiysk (2016 -)
Ilya Muromets (2017 -)
Aleksandr Sannikov (2018 - (planned); under construction)
Andrey Vilkitsky (2018 - (planned); under construction)
Ob (2018 - (planned); under construction)
Viktor Chernomyrdin (2019 - (current estimate); under construction)

As you can see Russia has won the ice breaking contest by the sheer volume of ships, and we could ask them to sell us a few since we are not making them anymore. And we could assign that newly purchased ex-Russian ice breaker to follow any US navy ships visiting cold water ports in the winter.