"asumir" is actually used in Spanish to mean "assume" (in the sense of "suppose").
Here we can read the following:
En el siglo xix amplía sus sentidos por influjo probable del inglés y en el español actual significa también ‘adoptar o adquirir [algo no material]’: «Manchester asumió una expresión de fastidio» (Otero Temporada [Cuba 1983]); «Su piel se asedaba, asumía una tersura vegetal» (Delibes Madera [Esp. 1987]); y ‘dar por sentado o por cierto’: «Los liberales piensan de otra forma, siempre y cuando asumamos que los liberales piensan» (Información [EE. UU.] 1996).
The definition above, where "asumir" is said to mean "dar por sentado" or "dar por cierto" (take for granted / regard as true), shows that the meaning (and use) of "asumir" in Spanish is not as extended as that of "assume" in English, where it can be used in different tenses and in any case where "suppose" is used (whenever "assume" refers to a mental process). Instead, it's not always safe to use "asumir" to mean "suppose", as "asumir" is more like "presuppose".
Thus, while in English we can say.
- I assume he's right. (meaning "I guess he's right")
in Spanish I would tend to say:
I would reserve "asumir" for cases where the supposition is used as a basis for some later course of action (that's why I say it is more readily associated with a premise), for example:
When I decided to follow his advice, I assumed he was right.
Cuando decidí seguir su consejo, supuse/asumí (= di por sentado) que tenía razón.
When this is the meaning that wants to be conveyed, "dar por sentado" or "dar por hecho" (take for granted or take as a fact) can be a safer way of saying "assume" in Spanish without falling back on the English cognate.
For further reference see Fundéu.