  Pulsars typically exhibit radio emission in the form of narrow pulses
originated from confined regions of their magnetospheres. A potential presence
of magnetospherically originated emission outside this region, the so-called
off-pulse emission, would challenge the existing theories. Detection of
significant off-pulse emission has been reported so far from only two pulsars,
B0525+21 and B2045-16, at 325 and 610 MHz. However, the nature of this newly
uncovered off-pulse emission remains unclear. To probe its origin we conducted
very high resolution radio observations of B0525+21 and B2045-16 with the
European VLBI Network (EVN) at 1.39 GHz. Whereas the pulsed emission is
detected at a level consistent with previous observations, we report absence of
any off-pulse emission above $42$ and $96\ \mathrm{\mu Jy\ beam^{-1}}$ (three
times the rms noise levels) for B0525+21 and B2045-16, respectively. Our
stringent upper limits imply the off-pulse emission to be less than $0.4$ and
$0.3\%$ of the period-averaged pulsed flux density, i.e., much fainter than the
previously suggested values of $1$-$10\%$. Since the EVN data are most
sensitive to extremely compact angular scales, our results suggest a
non-magnetospheric origin for the previously reported off-pulse emission.
Presence of extended emission that is resolved out on these milliarcsecond
scales still remains plausible. In this case, we constrain the emission to
arise from structures with sizes of $\sim (0.61$-$19) \times 10^3\ \mathrm{au}$
for B0525+21 and $\sim (0.48$-$8.3) \times 10^3\ \mathrm{au}$ for B2045-16.
These constraints might indicate that the two pulsars are accompanied by
compact bow-shock pulsar wind nebulae. Future observations probing intermediate
angular scales ($\sim 0.1$-$5\ \mathrm{arcsec}$) will help in clarifying the
actual origin of the off-pulse emission.
