GHz yielded the detection of a compact, unresolved component of 0.1 mJy centered at RA(J2000) = 12$^h$05$^m$31$^s$.7110 $\pm$ 0$^s$.0003, Dec.(J2000) = 50$^{\circ}$3246.729 $\pm$ 0.002 (@2011AN....332..379M). This provided an upper limit on the 5 GHz luminosity (assuming a radio spectral index $\alpha$ = 0.15), from which an upper limit on the ULX BH mass of $\sim10^{5} M_{\odot}$ was estimated using the Fundamental Plane of accretion (@2011AN....332..379M).

In this paper we present the first *Chandra* observations of the galaxy NGC4088 and the ULX N4088–X1, as well as quasi-simultaneous VLBI observations with the EVN at 5 GHz. We also report a reanalysis of the EVN data at 1.6 GHz, now imaged at the *Chandra* position reported in this paper, and the analysis of 23 *Swift* observations performed nearly at the same time (between April and September 2009) as the 1.6 GHz EVN observations (2009 June 1–2). With these data we investigate the proposed association between the X-ray and radio emission (for which a sub-arcsecond X-ray position is needed) of N4088–X1 and attempt to estimate the BH mass with the aim of revealing the nature of this ULX.

The paper is organized as follows: the observations and data analysis are presented in Section \[observations\], 