Here’s some relevant evidence.
John Loftis in English Renaissance Plays from the Spanish "Comedia" (1984) writes:
England and Spain were at war from 1585 until 1604, crucial years in
the development of their dramas; and travel between the countries was
all but impossible. Because of England’s alliance with the Dutch, who
had spent many years under Spanish rule, Englishmen had opportunities
to read Spanish plays and perhaps to see them in performance in
Amsterdam? The Anglo-Spanish dramatic relationship was one-sided: the
English knew something about the comedia; the Spanish knew nothing
about English drama. No Golden Age play of any consequence, so far as
is known, derives from an English one.
Loftis doesn’t believe in any future source-hunting progress. His arguments are about Spanish dramatic sources for English plays, but they could be applied vice-versa:
Hispanists of any nationality are likely to be familiar with
Shakespeare’s plays, and unlikely to overlook any probable source in
the comedia used by him.
So it’s not clear how Hispanists could overlook a Shakespeare source used by Lope, too. Loftis adds that all (rather obscure) English Renaissance plays that have sources in Spanish drama were “identified by late nineteenth-century Germans”, and there was no progress since that.
There’s also The Relations Between Spanish and English Literature (1910) by James Fitzmaurice-Kelly:
Meanwhile, to what extent was English literature known in Spain? Not
at all, so far as the external evidence goes. […] The material for La
Corona Tragica was obtained by Lope de Vega from George Conn, a Roman
Catholic divine long resident at Rome, whose Latin book on Mary Stuart
was published there in 1624. As Latin offers few difficulties to an
educated Spaniard, the fact that a contemporary Latin text was
translated into Spanish argues that it was expected to interest the
It is said that Shakespeare was almost unknown in Continental Europe before Voltaire published his Letters on the English – almost a century after Lope’s death. E.g. see Pujante, Ángel-Luis, “The French Influence on Early Shakespeare Reception in Spain: Three Cases of Unacknowledged Sources.” (2010) (PDF):
Shakespeare came to Spain in the 18th century, not directly from
England, but via France. He arrived as the monster created by
Voltaire, bringing with it the controversy about his vices and
Wikipedia article Reputation of William Shakespeare:
The knowledge of Shakespeare and his works in European countries, including Spain, arrived centuries after his death and not always
easily. Even if some folios containing Shakespearean plays managed to
arrive in Spain as soon as the end of the 16th century and the first
half of the 17th century, they did not have an impact on the theatre
and its audience.
Sur in English states that the oldest preserved edition of Shakespeare in Spain was published in 1632 (so it got there even later). Since Lope died in 1635, it’s unlikely he had a chance to encounter this book. And he probably didn’t know the language?
Soviet critic Plavskin (his article in the book “Шекспир в мировой литературе” (Shakespeare in World literature)) states that Lope didn’t know English. He doesn’t give a source, but he seems knowledgeable in the area. Plavskin also claims that Lope knew French, and his evidence for it comes from Lope’s autobiographical novel La Dorotea:
I began to collect books in every language and literature, for besides
acquiring the elements of Greek, I had much practice in Latin, learned
Tuscan well, and started French.
Perhaps his reasoning was this: Spaniards generally didn’t know English; there’s no evidence that Lope knew English; there’s some evidence that he didn’t know English at some stage of his life – we assume he didn’t know it.
If this is correct, it further lowers the probability of influence (doesn’t exclude an oral report as a source, though).