Sub-3:57 doesn't necessarily take sub-2:00 speed, but almost all of those who have done it had it and then some. Look at the all-time world and U.S. 1500 lists, then look at the runner's 800 PRs. I couldn't find a couple (**).
With the exception of Ayhan, almost all under 3:57 have run 1:58.xx or faster. Of the current crop of U.S. runners, only Pierce is close to that. She's also the only recent addition to the all-time U.S. performance list, as Slaney and Favor-Hamilton have nine of the top-10 marks.
All-time top-10
1. Tatyana Kazankina 3:52.47 (1:54.94)
2. Paula Ivan 3:53.96 (1:56.42)
3.Olga Dvirna 3:54.23 (1:56.90)
4. Haaisba Boulmerka 3:55.30 (1:58.72)
5. Sureyya Ayhan 3:55.60 (2:00.64)
6. Yulia Fomenko 3:55.68 (1:57.07)
7. Zamira Zaytseva 3:56.14 (**)
8. Marym Jamal 3:56.18 (1:57.89)
9. Yelena Soboleva 3.56.18 (1:57.28)
10,Tatyana Pozdnyakova 3:56.50 (**)
U.S. top-10
1. Mary Slaney 3:57.12 (1:56.90)
2. Suzy Favor-Hamilton 3:57.40 (1:58.10)
3. Anna Pierce 3:59.38 (1:58.80)
4. Jenny Barringer 3:59.90 (2:02.12)
5. Cristin Wurth-Thomas 3:59.98 (1:59.35)
6. Ruth Wysocki 4:00.18 (2:06.15)
7. Shannon Robury 4:00.33 (2:00.94)
8. Regina Jacobs 4:00.35 (1:58.08)
9. Diana Richburg 4:01.79 (**)
10. Jan Merrill 4:02.61 (**)